Irish-born Wogan had a successful television career, most notably anchoring Britain's flagship chat show during the 1980s, and hosted a hugely popular radio show for 27 years, retiring in 2009.
Prime Minister David Cameron said Britain had lost a huge talent.
"I grew up listening to him on the radio and watching him on TV," he wrote on Twitter. "His charm and wit always made me smile.
"Britain has lost a huge talent -- someone millions came to feel was their own special friend," he added.
Irish counterpart Enda Kenny said it was "with great sadness" that he learned of Wogan's death.
"Terry's humour and wit were unparallelled and he graced the top of his broadcasting profession for decades," he added.
"As an Irishman, Terry Wogan occupied a special place in British listeners' hearts and he acted in no small way as a bridge between Ireland and Britain."
- 'Subversive sense of fun' -
Wogan started his career as a banker in Ireland but after five years left to become a radio newsreader, beginning a remarkable broadcast career that was recognised with first an OBE and then a knighthood in 2005.
He began presenting the BBC Radio 2 breakfast show in 1972, helping it become Europe's most listened-to radio broadcast with a combination of smooth Irish charm and a subversive sense of fun.
He also hosted BBC TV chat show "Wogan" from 1982 until 1992, presiding over many memorable moments watched by millions of viewers.
They included football legend George Best turning up drunk, former sports presenter David Icke proclaiming himself as "a son of the Godhead", US actor Chevy Chase remaining silent throughout the interview and music icon David Bowie refusing to cooperate.
"David Bowie, well he probably wasn't at his best when I interviewed him," he later said.
"I thought a solid slap would have helped the situation. I didn't hit him, of course, but it came close."
- 'Most remarkable' Eurovision host -
Tony Hall, the BBC's director general, led the tributes, calling Wogan "a national treasure".
"For 50 years Sir Terry graced our screens and airwaves," he added. "His warmth, wit and geniality meant that for millions he was a part of the family."
Broadcaster and journalist Piers Morgan called him "one of the greatest broadcasters who ever lived".
Wogan offered commentary on the Eurovision Song Contest from 1980 until 2008, garnering a huge following with his light-hearted cynicism of the kitsch music competition.
In an address to Eurovision delegates shortly after handing over the British commentary to fellow Irishman Graham Norton, Wogan criticised countries who voted together along political lines in the annual singing contest.
He told them to stop taking the event so seriously because "everybody knows it is rubbish".
Despite occasionally irritating organisers with his remarks, for instance when he named the 2001 hosts "Dr Death and the Tooth Fairy", Eurovision on Sunday tweeted that it was "deeply saddened" by Wogan's passing.
"He was without doubt the most remarkable Eurovision commentator in history," the organisation said.
Wogan, born in Limerick in 1938, also hosted popular quiz show "Blankety Blank" and annual fundraising event "Children in Need" for over 30 years.
"Sir Terry has always been at the heart of the charity," said the fundraiser. "We will miss him so much."
He even entered the pop charts in 1978 with his version of popular English folk song "The Floral Dance".
A keen golfer, Wogan holds the distinction of sinking the longest ever televised putt, holing from 33 yards during a pro-celebrity tournament.
He is survived by three children, another one dying shortly after birth, and wife Helen Joyce.
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Prime Minister David Cameron said Britain had lost a huge talent.
"I grew up listening to him on the radio and watching him on TV," he wrote on Twitter. "His charm and wit always made me smile.
"Britain has lost a huge talent -- someone millions came to feel was their own special friend," he added.
Irish counterpart Enda Kenny said it was "with great sadness" that he learned of Wogan's death.
"Terry's humour and wit were unparallelled and he graced the top of his broadcasting profession for decades," he added.
"As an Irishman, Terry Wogan occupied a special place in British listeners' hearts and he acted in no small way as a bridge between Ireland and Britain."
- 'Subversive sense of fun' -
Wogan started his career as a banker in Ireland but after five years left to become a radio newsreader, beginning a remarkable broadcast career that was recognised with first an OBE and then a knighthood in 2005.
He began presenting the BBC Radio 2 breakfast show in 1972, helping it become Europe's most listened-to radio broadcast with a combination of smooth Irish charm and a subversive sense of fun.
He also hosted BBC TV chat show "Wogan" from 1982 until 1992, presiding over many memorable moments watched by millions of viewers.
They included football legend George Best turning up drunk, former sports presenter David Icke proclaiming himself as "a son of the Godhead", US actor Chevy Chase remaining silent throughout the interview and music icon David Bowie refusing to cooperate.
"David Bowie, well he probably wasn't at his best when I interviewed him," he later said.
"I thought a solid slap would have helped the situation. I didn't hit him, of course, but it came close."
- 'Most remarkable' Eurovision host -
Tony Hall, the BBC's director general, led the tributes, calling Wogan "a national treasure".
"For 50 years Sir Terry graced our screens and airwaves," he added. "His warmth, wit and geniality meant that for millions he was a part of the family."
Broadcaster and journalist Piers Morgan called him "one of the greatest broadcasters who ever lived".
Wogan offered commentary on the Eurovision Song Contest from 1980 until 2008, garnering a huge following with his light-hearted cynicism of the kitsch music competition.
In an address to Eurovision delegates shortly after handing over the British commentary to fellow Irishman Graham Norton, Wogan criticised countries who voted together along political lines in the annual singing contest.
He told them to stop taking the event so seriously because "everybody knows it is rubbish".
Despite occasionally irritating organisers with his remarks, for instance when he named the 2001 hosts "Dr Death and the Tooth Fairy", Eurovision on Sunday tweeted that it was "deeply saddened" by Wogan's passing.
"He was without doubt the most remarkable Eurovision commentator in history," the organisation said.
Wogan, born in Limerick in 1938, also hosted popular quiz show "Blankety Blank" and annual fundraising event "Children in Need" for over 30 years.
"Sir Terry has always been at the heart of the charity," said the fundraiser. "We will miss him so much."
He even entered the pop charts in 1978 with his version of popular English folk song "The Floral Dance".
A keen golfer, Wogan holds the distinction of sinking the longest ever televised putt, holing from 33 yards during a pro-celebrity tournament.
He is survived by three children, another one dying shortly after birth, and wife Helen Joyce.
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