The veteran BBC broadcaster, best known for his interview with disgraced US president Richard Nixon, was a star for more than 50 years.
His family said he died on Saturday night on board the Queen Elizabeth cruise ship, where he had been due to give a speech.
In a statement to BBC News, Sir David’s family said: “His family are devastated and ask for privacy at this difficult time.
“A family funeral will be held in the near future and details of a memorial service will be announced in due course.”
David Cameron sent his condolences on Twitter, saying: “My heart goes out to David Frost’s family. He could be – and certainly was with me – both a friend and a fearsome interviewer.”
My heart goes out to David Frost’s family. He could be – and certainly was with me – both a friend and a fearsome interviewer.
— David Cameron (@David_Cameron) September 1, 2013
Sir David probably interviewed more famous figure than any other broadcaster and won virtually every major television award.
The Cambridge Footlights alumnus first came to national prominence in the early 1960s with the groundbreaking satirical comedy show That Was The Week That Was.
Co-hosted by Frost, “TW3” showcased the work of up-and-coming writers and comedians and became a favourite with Saturday night audiences.
Sir David later set up his own company David Paradine Ltd which gave birth to many more hugely popular TV shows including Spitting Image and Through the Keyhole. He also helped start up LWT in 1967, and TV-am in February 1983.
He covered the moon landing in 1969 and his interview with Nixon, in which “Tricky Dicky” dramatically admitted he had “let down the country”, was at the time the most widely watched news interview in the history of television.
It was later dramatised into a sell-out West End play, and more recently the Hollywood movie Frost/Nixon.
Awful news about Sir David Frost. Was with him recently. In Broadcasting ,The Daddy of them all ! Such an influence on me and a true Gent.
— Eamonn Holmes (@EamonnHolmes) September 1, 2013
Other historic moments included the interview with Margaret Thatcher when Sir David introduced the word “bonkers” over the sinking of the Argentine warship the Belgrano during the Falklands conflict. She was furious.
Sir David’s interviewees included virtually every US president and British prime minister of the last fifty years, Prince Charles, Robert Kennedy, Henry Kissinger, Mikhail Gorbachev, Indira Gandhi, Yasser Arafat, Yitzhak Rabin, Nelson Mandela and many more.
Memorable interviewees from the world of entertainment included Orson Welles, Tennessee Williams, Noel Coward, Muhammad Ali, the Beatles, Clint Eastwood, Sir John Gielgud, Norman Mailer and others.
His Sunday morning interview programme Breakfast with Frost, famous for his trademark greeting, “Hello, good morning and welcome” ran on the BBC from January 1993 until May 2005.
As well as broadcasting, Sir David wrote 17 books, produced several films and started two television networks, London Weekend Television and TV-am.
In 1983, he married Lady Carina Fitzalan-Howard, second daughter of the Duke of Norfolk. They had three sons.
He was awarded an OBE in 1970 and received his knighthood in 1993.