Crow, 52, passed away in the early hours of Tuesday morning, the RMT said.
Crow has led the union since 2002 and been involved in several high profile disputes. As well as being the leader of the RMT, he was on the general council of the TUC
Crow's achievements included increasing RMT's membership from 57,000 in 2002 to more than 80,000 today.
London Overground Rail Operations Limited (LOROL) HR director Darren Hockaday worked closely with Crow over many years. He told HR magazine he was "devastated" to hear of his death.
"He was a great leader of his union and very successful at his job," Hockaday said. "I worked with him for six or seven years. We had a good working relationship and I'll miss working with him. He was always very interested in LOROL and he the company enjoyed working with him.
"I'd like to send my condolences to his family. I know his wife and daughter and my thoughts are with them and his wider family at this time. He was a very strong family man. That's one of the things we had in common."
Frank Douglas, former group HR director at Trasport for London, also paid tribute to Crow's effectiveness in representing his members.
"In spite of his public persona, he did what his membership paid him to do: get the best deal for them that he could," Douglas said.
"In performance management we focus on the individual meeting their objectives. In Bob's case you would objectively have to say, for his members, he always met or exceeded his."
Crow first became involved in politics in 1983 when he was voted as the local representative to the National Union of Railwaymen (NUR). In 1990, the NUR merged with another union to form the RMT, which he went on to lead from 2002 until his death.
Crow was often in confrontation with London Underground, calling a number of strikes, the most recent earlier this year. Crow was criticised for going to Brazil on holiday when negotiations to avert a strike were supposedly ongoing.
The RMT closed its offices today as a mark of respect.
*Photo by Jarle Vines
