Malaysian Prime Minister Mahathir Mohamad, on a visit to Tokyo, said "I hope that both parties will accept the fact that, in negotiation, both sides must be prepared to give in certain issues if they expect to reach a good conclusion."
At a train station in Seoul, the South Korean capital, people cheered and applauded as televisions screens broadcast the Trump-Kim handshake live.
"I really, really hope for a good outcome," said Yoon Ji, a professor at Sungshin University in Seoul. "I am hoping for denuclearisation and a peace agreement and also for North Korea's economy to open up."
Some questioned whether Kim would fully relinquish the weapons he may see as his only guarantee of survival.
"I am still not sure whether the North is willing to denuclearise or not," said Jo Han Won. "We never knew much about the North Korean regime and it's hard to distinguish what's true and what's false."
China, which provided a Boeing 747 that flew Kim from Pyongyang to Singapore, wants to ensure its interests are preserved in any negotiations, namely that they don't result in a unified Korea that is pro-American.
An editorial in the official English-language China Daily emphasised China's role in bringing Trump and Kim together, saying: "This would not only reward all those who have spared no efforts in their attempts to make their meeting a reality, it would also enable both to hail it as a success," the editorial read in part.
Hu Xijin, the editor of influential Communist Party newspaper Global Times, said that he was "truly happy for this moment". He dismissed as cynical those Chinese internet users who asked if Beijing had been marginalised in these talks.
"I think these people really grew up in negative energy," he wrote on his Sina Weibo microblog. Japan's largest newspaper, the Yomiuri, printed a special edition in both Japanese and English that was distributed for free in major cities 90 minutes after the meeting began.
- AP