By 2020, most women probably thought we would be a lot closer to equality in the workplace than we actually are. Yet, despite graduating from university in larger numbers than men for the past two decades and entering professions at the same rate, the number of women at the top remains stubbornly low.

Whether it is law firms or FTSE companies, film directors or politicians, the proportion of female leaders in the highest echelons sticks at about 5 per cent (there are currently six women leading FTSE 100 companies in the UK and only eight in the US top 100).

So to mark Sunday's International Women's Day, who better to ask what is going wrong than Sheryl Sandberg, the guru of female work empowerment, the woman who encouraged millions of us to Lean In, and COO of Facebook, the world's most powerful company.

"The reason we don't have as many female leaders is down to likeability," Sandberg says.

"The data shows conclusively that as men get more successful they are better liked by women and men — but as women get more successful they are less liked by both."

Sandberg may speak softly, a tiny package cosily wrapped in an overlarge peach cashmere jumper, red jeans and brown cowboy boots (she has just got engaged to a father-of-three who is taking her to country concerts), but make no mistake, she is as tough as chuck steak.

"The bias is subtle and insidious," she continues. "Let's review — we're coming out of college in higher numbers, so you can't argue based on educational achievement that girls aren't as smart or smarter . . . but in education you are taught to repeat something back. When you get to the workplace, leadership is much more important to performance. That, and being assertive. If women can't hold people to account without being seen as difficult or a nag, then as a woman you can't run teams effectively.

"I see it all the time," she says.

"To be perceived as a woman yelling, all you have to do is not say 'please'. For a man to be described as yelling, he'd have to scream and throw things. If a woman gives very specific instructions — which I do all the time — then you are controlling and difficult."

She sighs and dips a slice of apple into a mound of peanut butter. We are sitting in her sleek personal meeting room in the middle of the Facebook campus in Silicon Valley, but the girly snacks and confiding tone make the situation feel far more intimate.

"Women are called difficult all the time. And on top of that, people always assume we are doing things for our own purposes, whereas men are seen as leaderly and bold."

Sandberg is a mother of two, a boy aged 15 and a girl 12. How does she think we can shift these deeply ingrained prejudices for the next generation?

"We have to teach other people to understand the biases they have, and we have to teach women to get through it." She drops her already husky voice to a whisper: "We have to teach girls not to care quite as much [about what people think of them]. There is data to show that, at a leading business school, women who are married do much better than single women because part of the grade is oral participation, making strong arguments that other people agree or disagree with."

She swigs her coffee, then looks me straight in the eye. "If you are married, you are not trying to date within your class … you are not trying to be liked, so you are willing to make stronger points. And if you are single …" she sighs, raising her hands theatrically, "you are within the dating market in that class. So that shows if you can care less about being liked, you can be more assertive and get ahead. If you are the boss, you are going to make decisions people won't agree with and you are going to have to fire people, and they aren't going to like you then."

Given that for thousands of years being liked gave females an evolutionary advantage, whether in terms of attracting a mate or not getting attacked by physically stronger men, this is not as easy as it sounds. "Who wants to be disliked?" she nods. "[But] remember, it's not that men are trying to be more successful because they don't want to be liked, it works for them because they get more liked."

How do we change things? "Well, it starts in school …[teachers] writing report cards shouldn't say the little girl is bossy, but that she has executive leadership skills, she'll be a great CEO."

And it's not just teachers who have to change. "Parents have to have equal expectations for their children. There is a labour gap in the home — little girls do more domestic chores and get paid less." She also thinks we need more positive media messages around female ambition. "Where is the movie that ends with a girl getting a really great job, not a prince and a white horse?" I suggest Working Girl. "Oh yes, I love that movie, I'd forgotten that," she laughs.

The things holding women back go deep. Sandberg beams as she begins to tell me about her engagement to Tom Bernthal, CEO of a consulting firm and a former NBC producer, which happened the week before we meet. Tragically, her first husband, Dave Goldberg, died five years ago at the age of 47 after having a heart attack in a gym during a family holiday, leaving her a widow. This became the subject of her second book, Option B — so called because option A, being with Dave, was no longer possible, so she decided to "kick the shit out of option B". The book's synthesis of sharing her personal story alongside those of others, plus the best current research on grief and how to build resilience, all put together as a self-help manual for others coping with death, is classic Sandberg: putting her own heartache and hard-won experience to the service of others. Now she is using her new romance to try to help women shape their own destinies.

Sheryl Sandberg with her late husband David Goldberg. Photo / Getty Images

"When Tom and I were getting engaged, we planned the weekend together," she says, revealing that her new love, whom she met at a family bar mitzvah, is a great friend of her late husband's brother, Rob. "It was really important to me that we planned how we were going to get engaged together. We talked about that 'surprise me, propose' model, but this is one of the biggest decisions you are ever going to make." Sandberg shakes her head. "Why, in 2020, do we still have men surprising women, and women waiting for men to ask and not bringing up marriage themselves? I can't tell you how many conversations I have with the most fabulous young women who are, like, 'I'm not going to bring up marriage with my boyfriend.'" Sandberg strikes the table. "Why not bring it up if you want it?"

Tom is a father of three. Their combined family now contains three girls aged 12, 12 and 15. These days, Sandberg has to think about the example she sets, not only to her thousands of female followers across the world, but to the girls in her own family.

"We are talking about what we do to help women be more equal. We need equality in the home, in relationships and equality of voice, opportunity, pay! But it's all wrapped in together. One of the reasons women don't do as well in the workplace is because they are doing much more of the housework."

Women still suffer from that passive, Cinderella trait of waiting to be tapped on the shoulder and asked if they want something — whether that is for a promotion or a proposal. "So what do we do? We educate both sides. We educate women to ask, and then we have to change our institutions."

When I look sceptical, she recalls her time at Google — where she worked before moving to Facebook. There, "you had to apply for promotions. Far more men applied than women, but by sharing the data about how women weren't applying, more of them started to do so."

Sandberg is just as proactive in her personal life. She reveals that she told her brother-in-law that she fancied Tom, "and Rob fixed us up. He was the invisible hand behind it all. Rob talked to us both about what we were looking for and he thought we were a good match, and we got serious very quickly. Tom wasn't someone I'd met on a dating app, he was someone I knew — we'd been in the same rooms, I knew about his kids. He's unlike anyone I've ever met. He's amazingly giving, able to communicate really deeply about his feelings. He's one of the best fathers I've ever seen," she gushes.

His humour is another appealing trait. "He really makes me laugh, he's helped me lighten up." She decided he was the one when "I saw him dance at this bar mitzvah with his daughter, and it was amazing. And he is CEO of a company he founded but he is divorced, and on the days he has custody of his kids, he drives them to school and picks them up at 3pm. For a father to be that dedicated was a really big deal for me.

"When you have a death, the world doesn't give you permission to be happy," she continues. "I remember talking to adults who had lost a parent as children, and what they said was, 'I felt guilty if I was OK.' I saw my own kids do this, and I said, 'Daddy wouldn't want you to cry for ever.' " She gives full credit to her mother-in-law, who said, shortly after her husband died, " 'You are going to get married again and I will dance at your wedding.' That permission was so important."

Like her late husband, she is Jewish and was surprised to discover the Jewish mourning period for a spouse is only 30 days. "The rabbis wanted people to move on."

The rest of the world was not so understanding. She was viciously trolled when she announced she was dating again. She knows she has not been alone in that experience. "I have the data — women get judged much more harshly for dating post-bereavement."

She thinks successful women have a duty to speak up about the benefits of having power: "If you are the boss, you control your schedule." Recently she revealed that she leaves the office at 5.30pm to eat with her kids. Soon after, she was sent flowers by women from Facebook's internet rival Yahoo!, with a note saying: "Thanks, now we are going home to see our kids for dinner too." By talking about it, Sandberg had also given them permission to change things.

Despite selling upwards of 4.2 million copies worldwide, Lean In — her 2013 book about how women can better get on in the workforce — came in for its fair share of criticism. It was attacked for being a manual that helped only other privileged white women, and for its suggestion that it was up to females — many of whom were already leaning in so hard they were practically horizontal — to try harder, rather than blaming patriarchal institutions. Michelle Obama memorably criticised it with the words "that shit doesn't work all the time". Sandberg shrugs off the slights, but adds that in later editions she included more stories of women of colour and poorer folk, and that "in each country where it is published it now has a foreword by a local woman".

Women graduate from university at higher rates than men, but still remain under-represented at the top of the corporate world. Photo / Getty Images

While it is true that Sandberg is an extremely rich white woman (her personal net worth is estimated to be NZ$2.06bn), some of the Lean In lessons undoubtedly hit the spot with women around the world. There are now 46,000 Lean In "circles" encouraging women to support each other and own their ambition in countries as far spread as China, Chile and Cameroon. Coincidentally, on the flight to San Francisco, I sat next to a young South African woman who was on her way to Silicon Valley to take a job as finance director for a tech firm. We got chatting and she wrote a note for Sandberg, begging me to deliver it. It said: "Thank you from an anonymous admirer whose life you have profoundly influenced and shaped. Through your book you have taught me to keep my hand up, and your example has given me the courage and sense of self-worth to do it … I got my job while pregnant, and my first thought was, 'Right, I am going to handle it like a boss, like Sheryl did.'"

Such stories are legion on the Lean In website. Even the most cynical of us has to concede she made a difference. And how many other female business leaders have used their own social capital for such change? Most start talking about gender politics only on the way down.

So how did Sandberg make it through to the highest echelons when so many others didn't? For the first time she is a little evasive. There is an "umm" and she says, lamely, "Well, I guess I was lucky." Come on, I remonstrate, that is really unhelpful to other women. She laughs. "But it is also true. I always wanted to do something that mattered — I began my career working on leprosy in India, I have always chased impact."

She got a job at the World Bank straight after studying economics at Harvard, travelling to India to work on health projects including leprosy education. After returning to Harvard to study business, she became chief of staff to Larry Summers, Bill Clinton's deputy Treasury secretary (and her former mentor from her uni days), before working for Google, where she established its online advertising business. She joined Facebook in 2008 to bring her advertising smarts and human-management touch to Mark Zuckerberg's fledgeling social network. "I'm great at selling ads," she jokes later during the photoshoot, as she apologises for how stiff she becomes when a camera is pointed at her.

Her main job at Facebook was "growing the business", setting up the advertising sales operation all over the world that enables the company to micro-target ads against the interests that people reveal in their personal posts and likes. Having spent time in her company both at its HQ in California and in London, I can see exactly why she flourished. In a world of geeks happiest interacting in computer code, with her bouncy black blow-dry, queenly poise and incisive flattery, she is a beacon of warmth and empathy. I first met her at a Facebook round table in London, where I expressed scepticism about the company's record on everything from data privacy to failing to regulate harmful content. Instead of shutting me down, Sandberg congratulated me on my "great questions". Later, she agreed that I could come to California to interview her properly.

Sandberg has faced significant challenges in the recent history of Facebook. Photo / Getty Images

I spend a day at the Facebook campus on an industrial estate in Menlo Park, an hour's drive south from San Francisco. It feels like the US equivalent of Slough, both in terms of geography and glamour. I arrive early at the shiny blue "ideas factory" that stretches for miles, housing 40,000 workers next to the highway.

Despite its status as the centre of the tech world, Silicon Valley does not feel sophisticated. It's a small-town place where everyone agrees that tech is an unwavering force for good. The concentration of wealth is staggering. Tesla, Porsche and BMW dealerships line the highway, and the gossip is of tech titans so rich that they fly by private jet to ski on their own private mountains. The sense of a bubble is pervasive.

The Facebook factory looks like a hi-tech car plant, but has its own high street, with free restaurants, barbers and pop-up shops so workers never need to leave. Walking around, I can't help but feel conspicuous because of my gender. The engineering zone is a sea of identikit men — mostly of Asian descent— all wearing trainers and jeans. Up in the executive zone, where Sandberg and Zuckerberg sit at standard desks, like everyone else, in a huge open-plan office, the workers are overwhelmingly designer-clad white women. There is also a striking number of Brits, not least Nick Clegg, the former deputy prime minister, who now heads up Facebook's communications and lobbying team. With its nearly three billion global users, working for this company must make running a country of 66 million people like ours look like small beer.

There are posters on walls all around the plant reminding workers that "Nothing at Facebook is somebody else's problem", and encouraging empathy. Despite making $7bn of profit per quarter, all is not well. The company has gone from angel to devil. Once it boasted it was connecting the world. But in recent years it has been accused of failing to prevent rogue Russian agents using it to aid Donald Trump's election; of hosting a cesspit of illegal content; of providing a platform on which paedophiles operate with impunity; of exploiting our personal data to allow others to influence us; and even — by no lesser organisation than the UN — of abetting the genocide of the Rohingya Muslim minority in Burma, by allowing hate speech and propaganda to spread. How does it feel to be the subject of such universal criticism?

"What hurt about being in the middle of the Facebook storm," Sandberg responds smoothly, "was the things that we got wrong and the things that we need to do better — and in some ways I worry that all the good that social media does is just getting lost. And we have responsibility for that. No one was more upset about not understanding about Russian interference in the [2016 US presidential] election than we were. No one was more upset about Cambridge Analytica than we were," she says, referring to the British political consultancy, employed by the Trump campaign, that accessed the personal data of up to 87 million Facebook users, thus allowing precise and calculated targeting of voters, which some say swung the election.

Sandberg spends most of her time at the Facebook headquarters. Photo / Getty Images

Has she had some long dark nights of the soul about what Facebook has done? "It wasn't a long dark night of the soul because we understood what had happened, but there was definitely a shift afterwards in terms of my time and my priorities. I spend more time now on protection and community than I do on growth . . . It totally changed the structure of my job. The emphasis had always been on growth because we didn't see this kind of harm coming, we just didn't see it."

Is that an admission that there had been a "growth at any price" strategy before? In Burma, the platform went from having two million users in 2014 to 30 million by 2017 after it did a deal with the state-run telecom company to give subscribers free Facebook access. Yet despite that staggering growth, the company had no hate-speech detection algorithm in Burmese, and only a handful of moderators who spoke the language. So when content that fuelled the massacres of the Rohingya in 2017 was circulating freely on the site, Facebook effectively sat on its hands, even after campaigners repeatedly called for the content to be removed. (A UN report in 2018 stated that Facebook had played a "determining role" in stirring up hatred against the Rohingya.)

How does she feel about the UN warning Facebook that it could be abetting genocide? "Any time that anything goes wrong in our services it's awful, it's a massive shock, when people's whole world, their whole lives are involved it is terrible." She looks genuinely haunted.

What about the idea that Trump was elected because of Facebook? For only the second time in the interview, Sandberg has no fluent answer. "Umm. I don't want to answer that," she flounders. When I push, she adds: "Of course we are responsible for the data breach and we are taking responsibility."

Later, she clarifies: "Obama was the first Facebook candidate. Then people said Trump was the Facebook candidate. Whoever wins next will probably be called the Facebook candidate too. We offered support to the campaigns in 2016, including access to the same set of tools. The campaigns used Facebook differently, but they did a lot of other things differently too. No candidate is ever going to win by using Facebook well, while doing other things poorly."

While we are discussing the election, she suddenly says, "A lot of people have asked me why I stay . . . "

Why does she? She has made more than enough money to walk away. She gives me a soulful look. "I stay because I . . . am . . . so . . . passionate about what we do," she pauses, slowing her voice down. "Take the Option B groups [the bereavement groups she helped establish after her husband died]. They are all on Facebook. When I am having a bad day — when it's Dave's birthday and it's been five years, and even people I know really well are, like, 'You must be over it by now' — they understand. I believe in the power of that community. What is so hard is that all the good that Facebook does and social media does, I think is getting lost.

"There are 2.8 billion people using this service . . . there are always going to be some bad actors and our job is to stay ahead of them," she continues. "In the 40 minutes we've been sitting here talking, someone has found a blood donor on Facebook, someone has raised money for a charity and someone tried to do something bad. Of course, because of our scale that [last] one is big, so our job is to find that bad actor before anyone else sees it, and to get it down. But they will always try."

Is Facebook doing enough? It also owns Instagram, the photo-sharing app implicated in the death of Molly Russell, the British teenager who took her own life after viewing graphic posts about self-harm that the algorithm kept steering her towards? Shouldn't they be doing more to protect the vulnerable?

"We should and we are. We've always been proactive about protecting community standards. We don't allow nudity. We've always forced you to put your birth date in — a lot of other sites don't. If you put in the wrong birth date and we can triangulate and see that we think you are a kid, we'll throw you off. But I agree that we need to do everything we can to protect children." She asks if I have seen Messenger Kids, the platform and messaging app Facebook developed in 2017, used by children as young as six, which she says has "lots of protections" and gives "phenomenal control for parents". Most experts and child psychologists would argue that kids that young shouldn't be online at all.

"It's meant to be for tweens and teens, I don't know any six-year-olds who are using it," she says. Its name doesn't make that particularly clear to parents, and besides, a recent report found that 53% of children have a phone by the age of seven, with social media sites such as Instagram increasingly popular among girls aged seven to 16.

How much screen time does she allow her own children to have? Another "umm" and a pause. "We do have screen time limits, but I've never shared them publicly because the kids might mind. I am very protective of their privacy." She says she'll come back to me once she has chatted to them, but later emails me to ask if I would mind leaving the question out of this piece. Given that every parent I know worries about online time and the addictive features built into the products, I tell her that won't be possible. She comes back with a new answer: "We have strict screen limits during the week and very few screen limits over the weekend." Does she worry about her teenage daughter scrolling obsessively through idealised images on Instagram and the effect on her mental health?

"Sure, I worry they are going to get hit by a bicycle on their way to school and I worry that they will have a friend who is not nice and I worry about everything — what's the joke? Take the baby out, put the worry in," she smiles. Yes, but the specific concern I have raised is triggered by the company where she is COO, which has been accused of using algorithms that serve up harmful content to impressionable teens and younger. Yet another "umm" and a pause. "I definitely think that pressure to be a certain way for women and girls is a really big problem. We don't do bad diet ads anywhere, we have always been careful about that . . . and it is not only social media where that is a particular problem."

Yet there is a raft of studies that link the huge rise in anxiety in young women to social media use.

"I haven't seen the studies, so I won't comment." That strikes me as a cop-out. She then explains that she and her Lean In Foundation have done "a ton of work on media messages and advertising", and how important it is to use "women with normal body shapes". She has also created a whole new set of images with the Getty Foundation of different images of businesswomen.

"We need to change these images, but we need to change them throughout all media, not just online."

On the day we meet, Facebook is being accused of not paying its fair share of tax, and there are calls for the company to be broken up because its scale (it owns WhatsApp as well as Instagram) gives it a monopoly, along with Google, on selling targeted ads on our data. Given all the negative head winds, can she genuinely still believe that Facebook is a force for good?

This time there is no hesitation. "Yes, I do. Full stop. Yes. But not always. I think Facebook and social media and the internet do unbelievable amounts of good.

Also, a lot of harm has been done and it is our goal to maximise the good and to minimise the harm. I am worried that the harm is drowning out the good and we bear responsibility for that. I am so passionate about helping to lead this company through that hard time."

At this point, her assistant says firmly that the traffic is heavy and Sandberg has a plane to catch. But Sandberg isn't finished — and I doubt she's flying commercial. "I'm at the most important part of this interview," she beams. "Legislation in these areas is important, we are calling for it. GDPR [the EU law on data protection introduced in 2018] is a very good privacy regulation, the US should have national privacy legislation too."

On tax, she says: "We pay all we are legally obliged to." Many would argue the company should be obliged to pay much, much more. And, to be fair, it has said it will. On the 2020 US election, she says Facebook already has some of its top hackers thinking up what "the Russians or others could be doing to disrupt the process". Whatever the result, she thinks Facebook will still be deemed accountable in some way.

I think that for a woman who cares so much about her public image and being seen as a force for good, she must have found the attacks on Facebook incredibly hard. As if reading my mind, she pauses and says:

"It's hard, but it should be hard for Mark and it should be hard for me. You asked if we have late-night soul searching. Of course we have the soul searching, but I would say it is more like late-night action and early morning action around what can we do better. I don't think it took us a lot of time between seeing [the problem] and understanding it and taking action to try to fix the problem. For instance, we now run red team drills all the time. Like, 'Pretend you were a bad guy, what would you do now?' We are investing billions of dollars — and we should be — in stopping this. But, for me, I feel like I am fighting to protect and continue something I know does so much good, particularly for women who want to start their own businesses."

She makes the point that social media has also given a voice to women around the world who previously didn't feel heard. "In the old world, only a very small number of people had a voice. But now, me as a high-school girl in Miami, I would be able to speak." Or, I suggest, Malala, who raised the plight of girls' education after being shot by the Taliban for going to school in Pakistan, or Greta Thunberg, who started the School Strike for Climate that grew online, or the thousands of women who took part in the #MeToo movement. For them, social media has really helped to change the world.

At one time, it was rumoured Sandberg might go into politics herself. It was said that if Hillary Clinton had won, Sandberg would have been Treasury secretary.

Is that true? "Nope, I never was and I'm not going to."

That's a shame. The world needs more powerful women — and nothing will change until more of us step up and speak out.

- The Times