His first title was at middleweight, the next (amazingly, given his relatively small size) at heavyweight and the light heavyweight title came later (when he was 40) when the division was first recognised.
Other boxers have made the move from light heavyweight champion to heavyweight titleholder but Fitzsimmons is the only man to have succeeded in winning titles in the opposite sequence.
Fitzsimmons was a colourful character – one of his more notable fights was controversially refereed by the notorious gunslinger, Wyatt Earp, complete with pistol strapped to his thigh - and he made his reputation in a series of fights in the United States.
His most famous fight took place in 1897, in Carson City, Nevada, and was against the heavyweight titleholder "Gentleman" Jim Corbett.
The fight, which was one of the earliest sporting contests to be filmed (and the film can still be seen), is remembered both for the manner of Fitzsimmon's victory and for the role played by Fitzsimmon's wife, Rose.
The much heavier Corbett was a worthy champion and was boxing well; he looked to be on course to retain his title.
But, late in the fight, Rose – who was ringside – famously called to her husband, "Hit him in the slats, Bob!"
Rose had seen that Fitzsimmons needed to switch his attack from his opponent's head to the body.
Fitzsimmons duly followed his wife's advice, came in under Corbett's lead, and unleashed his famous "solar plexus" punch.
The punch was so fearsome that Corbett went down and he was so disabled by its power that he was unable to continue.
Fitzsimmons (and, one presumes, his wife as well) did not always find it easy to enjoy the fruits of his success.
He spent unwisely, was addicted to gambling and was unduly susceptible to confidence tricksters.
But his achievement lives on as one of the great moments in boxing – and Rose's injunction to "hit him in the slats" as one of the most perceptive and decisive interjections ever offered at a critical moment in a sporting arena.
Anthony Joshua proved himself, in defending his title against Joseph Parker, a more formidable opponent than "Gentleman" Jim had been when fighting Bob Fitzsimmons.
But Joseph Parker stayed on his feet and carried with him, one feels, the spirit of Bob Fitzsimmons – all that was missing was Rose.