A family statement said he was aboard the Queen Elizabeth on Saturday night where he was to give a speech.
Sir David’s career spanned journalism, comedy writing and daytime television presenting, including The Frost Report.
Internationally, he will be remembered for his revealing interviews with former US President Richard Nixon.
A statement 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.”
The BBC’s Barney Jones edited his Breakfast with Frost programme on the BBC for more than 10 years.
Of his friend and colleague, he said: “David loved broadcasting, did it brilliantly for more than 50 years and was eagerly looking forward to a host of projects – including interviewing the prime minister next week – before his sudden and tragic death. We will all miss him enormously.”
Former US president Richard Nixon: “I let my country down”
Prime Minister David Cameron said: “Sir David was an extraordinary man, with charm, wit, talent, intelligence and warmth in equal measure.
“He made a huge impact on television and politics.”
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Born in Kent, Sir David studied at Cambridge University where he became secretary of the Footlights club, and met future comedy greats such as Peter Cook, Graham Chapman and John Bird.
After university he went to work at ITV before he was asked to front the BBC programme That Was The Week That Was, which ran between 1962 and 1963.
Casting a satirical eye over the week’s news, the show boasted scriptwriters including John Cleese, John Betjeman and Dennis Potter.
The Frost Report brought together John Cleese, Ronnie Barker and Ronnie Corbett in a sketch show which would influence many comedy writers including the Monty Python crew.
Sir David’s often-mimicked catchphrase “hello, good evening and welcome” was by now in full use.
One of The Frost Report’s most enduring pieces was the “class sketch”, featuring Cleese, Barker and Corbett.
The Frost Programme for ITV followed, which saw Sir David move away from comedy into in-depth interviews with political figures, royalty and celebrities.
It was on this programme that he had a terse interview with then prime minister Margaret Thatcher over the sinking of the Argentine cruiser Belgrano during the Falklands conflict.
At the same time, Sir David began work on The David Frost Show in the US.
He later conducted a series of interviews with Mr Nixon, who had resigned the presidency two years earlier, in which the former president came close to apologising to the public for his role in the Watergate scandal.
Their exchanges were eventually made into the film Frost/Nixon – based on a play – which saw Michael Sheen portray Sir David Frost to Frank Langella’s Nixon. Sir David himself appeared at the premiere of the film in 2008.
Playwright Peter Morgan said: “He was a legendary broadcasting figure and a member of the British broadcasting landscape for two generations and in many ways his success was very un-English.
“He was a pioneer. He combined being a satirist and someone who one satirised. It was an extraordinary, four-dimensional, vivid career…and he was a great lunch.”
‘In love with television’
In the 1990s, Sir David presented Through the Keyhole, which he also produced, alongside Loyd Grossman.
The show saw Grossman take a tour round the home of a celebrity while a panel of guests tried to guess “who lives in a house like this”.
Paying tribute to his friend, Grossman said: “He so effortlessly roamed all across the piste… from comedy to current affairs to light entertainment for 50 years.
“Yet in his presence you forgot you were dealing with the Leviathan of broadcasting and just thought here is a wonderful man, generous, enthusiastic and always excited. He was in love with television.”
In 1993, the year he was knighted, he began presenting Breakfast with Frost – which had begun life on ITV – a Sunday show on BBC in which he interviewed newsworthy figures.
BBC director general Tony Hall said: “From satire to comedy to the big political interviews, for more than 50 years he brought us the history of the world we live in today, that’s the mark of the man.”
Sir David joined broadcaster Al-Jazeera in 2006 when it launched its English-speaking service.
He married his second wife, Lady Carina Fitzalan-Howard, in 1983 and they had three sons.
‘Learnt from David Frost’
He worked closely with a number of charities over the years, including Alzheimer’s Research Trust, the Motor Neurone Disease Association and health charity Wellbeing of Women.
The latter’s chairman, Sir Victor Blank, said: “It’s a sad day and David’s tremendous contribution over the last half century to television will be honoured.
“David was also a marvellous husband, father and friend, but not often recognised is the time, generosity and support he gave to so many charities, not least the 25 years he has spent helping Wellbeing of Women.”
And the BBC’s Andrew Marr added: “David Frost changed British broadcasting not once but twice. He was a prime mover in the satire boom of the 1960s. A lot of that was down to him and his drive and shaping, influence and personality.
“And then he changed the whole style of political interviewing, what could be said, how it was done, the whole approach. And I think today there are two types of political interviewer – those who’ve learnt from David Frost, and second rate interviewers.”