Friends and colleagues, both in Ireland and in the UK, praised the Limerick native, whose career spanned six decades.
A statement from the Wogan family said Terry had died “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,” the Wogan family statement said.
BBC director general Tony Hall described Mr Wogan as a “national treasure”, whose “warmth, wit and geniality meant that for millions he was a part of the family”.
“Today, we’ve lost a wonderful friend. He was a lovely, lovely man and our thoughts are with his wife and family,” he said.
Last November, Mr Wogan cancelled his scheduled appearance on the BBC’s annual Children In Need appeal, due to ill-health, the first time he did not host the fundraising show in its 35-year history.
“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,” Mr Hall said.
Both Taoiseach Enda Kenny and British Prime Minister David Cameron acknowledged Mr Wogan’s passing.
“His always entertaining, and often unforgiving, commentary of the Eurovision Song Contest provided viewers here, and in Britain, with endless entertainment,” Mr Kenny said.
Mr Wogan finished his long-running stint presenting BBC Radio 2’s breakfast show in 2009, with Chris Evans taking over the high-rating slot.
“We are all so terribly sad, upon hearing of the passing of Terry,” Mr Evans said.
“I can’t put into words how the whole Radio 2 family is feeling. To many of us, Terry was Radio 2. We still can’t believe it,” he said.
Signing off on his last Wake Up To Wogan, in 2009, the host admitted that it was a day he was ‘dreading’.
“It’s always been a source of enormous pride to me that you have come together in my name, that you are proud to call yourself my listeners, that you think of me as a friend, someone that you are close enough to laugh with, to poke fun at and, occasionally, when the world seemed just a little too cruel, to shed a tear with,” he said at the time.
“And the years together with you have not only been a pleasure, but a privilege. You have allowed me to share your lives with you. When you tell me how important I have been in your lives, it’s very moving. You have been every bit as important in mine.
“I am not going to pretend that this is not a sad day — you can probably hear it in my voice. I am going to miss the laughter and the fun of our mornings together,” he said.
“Thank you, thank you for being my friend.”