On the rock'n'roll scene since the age of 15, Keys played with Buddy Holly, John Lennon and Eric Clapton.
The 70-year-old was most famous for his work with the Stones on the classic track Brown Sugar and their album Exile On Main Street.
The band issued a statement mourning "the loss of their very dear friend and legendary saxophone player".
The statement went on: "Bobby made a unique musical contribution to the band since the 1960s. He will be greatly missed."
Guitarist Keith Richards said: "I have lost the largest pal in the world and I can't express the sense of sadness I feel although Bobby would tell me to cheer up."
Keys featured in Richards' autobiography Life in 2010 and recalled the first time he met the Stones in the mid-1960s.
Like Buddy Holly, Keys was born in the Texas town of Lubbock and had toured with him and his band The Crickets in the 1950s.
But he was not impressed that the Stones had recorded a cover of Buddy Holly's Not Fade Away.
Keys recalled the moment: "I said 'Hey, that was Buddy's song. Who are these pasty-faced, funny-talking, skinny-legged guys to come over here and cash in on Buddy's song?'''
Keys played during the band's Glastonbury headline slot in 2013, although in October he was forced to pull out of dates in New Zealand and Australia due to poor health.