Remembered chiefly for his work as an athletics commentator, Coleman worked at the BBC for almost 50 years, covering 11 Olympic Games and six football World Cups.
"We regret to announce the death of David Coleman OBE. After a short illness he died peacefully with his family at his bedside," his family said in a short statement.
The BBC's Director-General, Tony Hall, said: "David Coleman was one of this country's greatest and most respected broadcasters. Generations grew up listening to his distinctive and knowledgeable commentary.
Coleman joined the BBC in 1954 and was affectionately known for on-air blunders that became known as 'Colemanballs'.
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He was made an Officer of the Most Excellent Order of the British Empire (OBE) by Queen Elizabeth II for services to broadcasting in 1992 and retired in 2000.
In the year of his retirement, he became the first broadcaster to receive an Olympic Order medal to recognise his contribution to the Olympic movement.
Gary Lineker, the former England striker who now works as a BBC presenter, wrote on Twitter: "Sad to hear, David Coleman has died. A giant of sports broadcasting. Brilliant, gifted, precise and concise. Much more than 'one-nil'.