The Rail, Maritime and Transport union made the announcement "with the deepest regret".
A brief statement said: "It is with the deepest regret that RMT has to confirm that our general secretary Bob Crow sadly passed away in the early hours of this morning.
"RMT would request that all media respect the privacy of the friends and family of Bob Crow at this distressing time."
Mr Crow, who was 52, was one of the most high-profile, left-wing union leaders of his generation, sparking as much anger from passengers hit by rail and Tube strikes, as praise from his members for winning pay rises.
He was constantly involved in industrial disputes and campaigns and led a walkout by London Underground workers last month in a dispute over ticket office closures.
The straight-talking south Londoner was a passionate supporter of Millwall Football Club.
His death caused shockwaves in the trade union movement today.
Manuel Cortes, leader of the TSSA rail union, who stood on picket lines with Mr Crow during last month's Tube strike, said: "Bob Crow was admired by his members and feared by employers, which is exactly how he liked it.
"It was a privilege to campaign and fight alongside him because he never gave an inch."
Under Mr Crow's leadership, membership of the RMT increased by more than 20,000 to 80,000, embracing workers ranging from seafarers and rail staff to cleaners.
He spoke at rallies and meetings most weekends, and was always in demand to support campaigns.
He became a target of right-wing commentators, who criticised his militancy and involvement in disputes.
He even had to explain why he had gone on holiday in the run-up to last month's London Underground strike.
Former Mayor of London Ken Livingstone led tributes to Crow saying he fought for his members despite being demonised by the right wing press.
Speaking of his shock at the news, he said: "I assumed he would be at my funeral not me at his."
He said: "He fought really hard for his members. The only working-class people who still have well-paid jobs in London are his members."
He said Mr Crow was "broadly right on most key issues" and that if more people had fought for the conditions of the working classes "this country would be a much better place."
"With the passage of time people will come to see that people like Bob Crow did a very good job."
UK Independence Party (Ukip) leader Nigel Farage tweeted: "Sad at the death of Bob Crow. I liked him and he also realised working-class people were having their chances damaged by the EU."
Mr Crow had been campaigning for the No2EU political party in May's European elections, arguing that workers were suffering because of policies from Europe.
In a statement, 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.
"Bob fought tirelessly for his beliefs and for his members.
"There can be absolutely no doubt that he played a big part in the success of the Tube, and he shared my goal to make transport in London an even greater success.
"It's a sad day."
Sir Peter Hendy, London's Transport Commissioner, said: "We are shocked by this terribly sad and unexpected news. Our thoughts are with Bob Crow's family, friends and all those he represented."
Grahame Smith, Scottish TUC general secretary, said: "Everyone at STUC is deeply saddened by the passing of Bob Crow. Since 2002, Bob led his union with courage and distinction and will be greatly missed, not just by his own members but by trade unionists across Scotland and the whole of Britain.
"STUC echoes the request of RMT that everyone should respect the privacy of the friends and family of Bob Crow at this distressing time."
Former mayor of London Ken Livingstone said Mr Crow fought for his members despite being demonised by the right wing press.
"He fought really hard for his members. The only working-class people who still have well-paid jobs in London are his members."
Labour leader Ed Miliband said: "Bob Crow was a major figure in the labour movement and was loved and deeply respected by his members.
"I didn't always agree with him politically but I always respected his tireless commitment to fighting for the men and women in his union. He did what he was elected to do, was not afraid of controversy and was always out supporting his members across the country.
"He was a passionate defender of and campaigner for safe, affordable public transport and was a lifelong anti-fascist activist.
"My thoughts are with his family, friends and colleagues in the RMT and wider union movement at this difficult time."
Mr Crow's older brother, Richard Crow, said he believed the union leader had suffered a heart attack in the early hours of today.
Speaking to Sky News, he paid tribute to the "loveable little rogue".
Richard, whose father was also a trade union leader, said: "It's very sad.
"It was about 7am that I got the call (from my sister). I presume some time in the night he had some problems. We're really trying to find out exactly what happened.
"We grew up together in Chigwell in Essex and he was a very likeable chap - no matter what people said about his politics.
"He was honest, he looked after the people he was supposed to look after, and he was a great man as far as honesty and beliefs went.
"He was a person who believed in justice."
Richard, who is three years older than his brother, said: "People moaned that he lived in a council house, that he never drove a car - he lived a life of the average guy in the street and that's a rare thing these days.
"When people have a high office in life they fall for the big trappings of the flash cars and the big hotels and big houses. But Bob wasn't like that, he was a genuine person of the people."
Remembering their childhood, Richard said: "He was an annoying little rascal to me when I was a kid.
"He'd forever be blackmailing me that he would tell Mum and Dad the sort of things that I got up to.
"But he was one of those loveable little rogues, one of those guys that had bundles of friends.
"He would be a cheeky chappie Cockney kid. He was a lovely kid to grow up with.
"I can't say anything bad about him as far as the person goes.
"Politically we were completely different, but growing up he was a typical London scallywag."
Mr Crow also described his Millwall FC-supporting brother's passion for sport, particularly enjoying cricket and football.
Scottish politicians joined those paying tribute to Mr Crow.
First Minister Alex Salmond said: "Bob Crow was a formidable and tireless champion for the workers and causes that he represented. His loss will be felt, not just by the RMT, but by all trade unionists, who have lost a fearless and effective campaigner.
"It is clear from the tributes we have heard this morning that, even amongst those who did not agree with Bob Crow's politics, that there was genuine respect for the principled way he spoke out for his members."
Scottish Labour Leader Johann Lamont said: "Bob would take on anyone when he thought it was in the interests of working men and women and the confidence and certainty in which he made his arguments had to be admired.
"He had become a giant figure in the trade union movement and will be sadly missed by the many trade union members who held him in high esteem."