The union confirmed in a statement on its website that Crow died in the early hours of Tuesday morning.
It is believed he suffered a heart attack.
It added: “The union’s offices will be closed for the rest of the day and the union will make further announcements in due course.”
Crow was a tireless campaigner during his years as general secretary to improve the pay and conditions for rail and underground maintenance workers.
He was a hugely controversial figure in London after leading a series of tube strikes.
London Mayor Boris Johnson said: “I’m shocked. Bob Crow was a fighter and a man of character.
“Whatever our political differences, and there were many, this is tragic news.”
Transport Secretary Patrick McLoughlin said: “Like many, I will remember him as a passionate voice for safety on the railways and the wellbeing of those who work on them.
“While we may not always have agreed on how to run our railways, he was a powerful advocate who led his organisation from the front and made an important contribution to the debate around the future of rail services in this country.”