Edinburgh-born Robb started his broadcasting career at Northsound Radio in Aberdeen before moving to Radio 1.
After a stint at Radio Scotland, he joined Five Live in 1994, which was where he would make his name as a daytime presenter, most recently on the midday news.
During his time at the station, Robb covered a range of major stories, including the death of Princess Diana and London winning the 2012 Olympics. He also secured an exclusive interview with Nelson Mandela.
Expressing his sadness at Robb's passing, Five Live controller Adrian Van Klaveren said: "Allan had a quiet authority and warmth to his broadcasting that listeners loved - he was a great professional and is a great loss to the radio industry."
Five Live breakfast presenter Nicky Campbell knew Robb for over 40 years after the pair went to the same school in Edinburgh.
"He was a great broadcaster and professional and had a quick and agile mind, asking questions that people couldn't hide from," said Campbell.
"It's one of the saddest days of my life. He pointed up to Broadcasting House and said, 'We work there for the BBC'. He never forgot that."