The veteran broadcaster, known for his velvety voice on radio and television, was one of the UK and Ireland’s best known stars.
A statement said Limerick-born Sir Terry died surrounded by his family.
Tributes have poured in from a host of stars, with Prime Minister David Cameron saying Sir Terry was “someone millions came to feel was their own special friend”.
Sir Terry was last on air on BBC Radio 2 just under three months ago, on Sunday November 8, and days later was forced to pull out of presenting Children In Need at the last minute due to health issues.
A family statement issued by the BBC said: “Sir Terry Wogan died today after a short but brave battle with cancer. He passed away surrounded by his family. While we understand he will be missed by many, the family ask that their privacy is respected at this time.”
BBC Director General Tony Hall described Sir Terry as a “national treasure”.
He said: “Terry truly was a national treasure. Today we’ve lost a wonderful friend. He was a lovely, lovely man and our thoughts are with his wife and family.
“For 50 years Sir Terry graced our screens and airwaves. His warmth, wit and geniality meant that for millions he was a part of the family.
“Wake Up To Wogan was for millions of Radio 2 listeners the very best way to start the day. For decades he’s been such a huge part of the BBC on television and radio and leaves so many wonderful memories.
“At the centre of Children In Need since its beginning he raised hundreds of millions of pounds and changed so many lives for the better. He leaves a remarkable legacy.”
Prime Minister David Cameron tweeted: “My thoughts are with Terry Wogan’s family. Britain has lost a huge talent – someone millions came to feel was their own special friend.
“I grew up listening to him on the radio and watching him on TV. His charm and wit always made me smile.”
Helen Boaden, director at BBC Radio, said: “Sir Terry was a radio legend. For decades, he gave great pleasure to radio listeners with his wit, warmth and inimitable humour. He was an extraordinary broadcaster but also incredibly good fun, and will be sorely missed.”
Bob Shennan, controller at Radio 2, said: “As the host of Wake Up To Wogan, Terry established himself as one of the greatest and most popular radio hosts this country has ever heard.
“We were brightened by his wonderful personality and charm as he woke us up every weekday morning, becoming an essential and much-loved part of our lives.
“His millions of listeners adored him, as did his whole Radio 2 family. We will miss him enormously and our thoughts at this very sad time are with Helen and all the family.”
Paying tribute to his friend, BBC broadcaster Jeremy Vine said: “Terry started doing the Radio 2 breakfast show when I was six. When, aged 37, I joined the network, he was unfailingly encouraging and friendly. He did nearly 40 years at breakfast, with an intermission for TV work: surely an unbeatable record.
“Someone asked Terry how many listeners he had. Instead of answering nine million, which would have been accurate, he said: ‘Only one.’
“And it was this approach that made him one of the greatest broadcasters this country has ever seen. He only ever spoke to one person.”
Vine also quoted a conversation between Sir Terry and the Queen, during which she asked him how long he had worked at the BBC.
Sir Terry replied: “Your Majesty, I’ve never worked here.”
More tributes have poured in from the world of showbusiness and beyond.
Former BBC Radio 2 colleague Jeremy Vine said on Twitter: “Truly one of the greatest we have ever seen. And treated his colleagues like friends.”
Tony Blackburn added: “I can hardly believe my old friend Sir Terry Wogan has died. RIP Terry and thanks for being a friend.”
Fellow broadcasting DJ Lauren Laverne wrote: “Oh, man. RIP Sir Terry Wogan. Broadcasting hero”, while Julian Clary said: “RIP Terry Wogan. A witty, warm and wonderful man.”
Richard Osman, of TV’s Pointless, said: “So sad about Sir Terry Wogan. What a wonderful, clever, funny man. Did nothing but make the world a happier place.”
Cricketer Ian Botham wrote: “Very sad to hear the news about Sir Terry Wogan passing away this morning !! A wonderful man just the same rain or shine !! RIP”, while actor Sir Roger Moore added: “Oh no. It’s a sad weekend. BBC News – Sir Terry Wogan: Veteran broadcaster dies aged 77.”
Fellow presenters spoke of their admiration for Sir Terry’s extraordinary career and broadcasting abilities.
Piers Morgan said: “RIP Sir Terry Wogan. One of the greatest broadcasters who ever lived. Such sad news.
“There are great TV broadcasters & great radio broadcasters. Terry Wogan was the greatest TV & radio broadcaster. A legend of all airwaves.”
He also tweeted a quote from Sir Terry: “‘People who are successful should never forget that it’s 90% luck.”‘
Graham Norton said: “He made it seem effortless and for a young boy in Ireland he made it seem possible. RIP Sir Terry Wogan. I’ll raise a glass during song 9.”
Clare Balding wrote: “Devastated about Terry Wogan. Grew up listening to him & have been privileged to know & work with him. Warm, generous and supremely talented.”
Noel Edmonds said: “He mastered broadcasting’s greatest challenge – to be admired and respected by public and peers alike – a true master has gone.”
Gaby Roslin added: “No no no! In total shock. Goodbye my cheeky chum. Forever in our hearts. RIP Terry Wogan.”
Dan Walker said: “Such sad news about Sir Terry Wogan. Had an amazing ability to make each one of his millions of listeners feel it was just him and them.”
Greg James wrote: “Warmth on radio is so underrated and Terry Wogan was the master of it. Add to that insanely funny and irreverent and you have a true great.”
Davina McCall added: “Terribly sad news about Terry Wogan. My thoughts are with his family. X RIP.”
Actor Matt Lucas said: “So so sad to hear the news today about Terry Wogan. Warm, traditional and yet brilliantly subversive. A true TV and radio icon. What a loss.”
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