In a short statement his family said his death, at his home in Surrey, was “unexpected but peaceful”.
It read: "It is with great sadness we announce the passing of golfing and broadcast legend Peter Alliss. Peter was a devoted husband, father and grandfather and his family ask for privacy at this difficult time."
European Tour chief executive Keith Pelley said: "We are deeply saddened to hear of the passing of Peter Alliss, truly one of golf's greats.
"Peter made an indelible mark on everything he did in our game, but especially as a player and a broadcaster, and he leaves a remarkable legacy. Our thoughts are with his wife Jackie and the Alliss family."
Born on February 28, 1931, in Berlin, where his father Percy worked as a club pro, Alliss followed in his father's footsteps and quit school at the age of 14 to work for him at Ferndown Golf Club in Dorset, before turning professional himself two years later.
After his career was largely put on hold by two years of National Service in the RAF from 1949 to 1951, Alliss soon began to make a name for himself and finished ninth in the 1953 Open, one of five top-10 finishes in the event.
That performance helped Alliss gain selection for the Ryder Cup in October, making he and Percy the first father and son to play in the biennial contest, but Peter suffered a crucial singles defeat to Jim Turnesa as the home side lost by a point at Wentworth.
His move into broadcasting came about after he was overheard by the BBC's Ray Lakeland talking to a friend on a flight back from a tournament in Ireland in 1960.
Alliss combined commentary stints at the following year's Open Championship at Royal Birkdale with finishing eighth behind Arnold Palmer, helping to raise his profile to such an extent that he was chosen to give Sean Connery golf lessons before the actor played James Bond in the 1964 film 'Goldfinger'.
Alliss became the BBC's lead commentator in 1978 and was due to celebrate his 60th year in broadcasting in 2021. In November he commentated on the Masters from home due to the coronavirus pandemic and his own failing health.
The director of BBC Sport, Barbara Slater, said: "Peter was the voice of golf. He was an absolute master of his craft with a unique ability to capture a moment with a magical turn of phrase that no one else could match.
"He transcended his sport as one of the greatest broadcasters of his generation. He will be terribly missed and our thoughts are with his family, friends and colleagues at this very sad time."
The BBC director general, Tim Davie, added: "No one told the story of golf quite like Peter Alliss. He captured golf's drama with insight, wisdom, and humanity. He was a legendary commentator who brought the game to life for millions of us."
Additional reporting by PA
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