He passed away after a short illness, his family have confirmed.
In a statement, they 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 said: "Terry truly was a national treasure."
The popular TV presenter enjoyed a career spanning 50 years, including hosting the Eurovision Song Contest and Children In Need.
However, he was forced to pull out of hosting last year due to ill health.
Chris Evans, who took over from Terry after he left BBC Radio 2, has paid tribute to the veteran broadcaster, saying: "We are all so terribly sad upon haring of the passing of terry. I can't put into words how the whole Radio 2 family is feeling.
"Our most heartfelt thoughts go out to Helen, Mark, Alan, Katherine and Vanessa. To many of us Terry was Radio 2. We still can't believe it."
RIP Terry. You'll be missed.
Read some of the celebrity tributes below:
What incredibly sad news about Sir Terry Wogan. I absolutely loved him on the radio.— Matt Edmondson (@MattEdmondson) January 31, 2016
He has now left, entirely empty, my childhood kitchen.— Alison Moyet (@AlisonMoyet) January 31, 2016
The class of '77 have truly grown.
I shall miss your voice terribly.#TerryWogan
So sad to hear Terry Wogan has passed. Such a lovely man. We had the honour of working with him many times. You'll be missed Sir. The boys X— The Overtones (@the_overtones) January 31, 2016
'People who are successful should never forget that it's 90% luck.' - Terry Wogan.— Piers Morgan (@piersmorgan) January 31, 2016
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— Greg James (@gregjames) January 31, 2016
Omg RIP Terry Wogan— Luisa Zissman (@TheLuluLife) January 31, 2016
So sad to hear the passing of Sir Terry Wogan— Ben Fogle (@Benfogle) January 31, 2016
Devastated about Terry Wogan. Grew up listening to him & have been privileged to know & work with him. Warm, generous & supremely talented— Clare Balding (@clarebalding) January 31, 2016
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.— Matt Lucas (@RealMattLucas) January 31, 2016
Such sad news about Sir Terry Wogan. Television Jedi— Leigh Francis (@LeighFrancis) January 31, 2016
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.— graham norton (@grahnort) January 31, 2016
Terribly sad new about Terry Wogan . My thoughts are with his family. X RIP— Davina McCall (@ThisisDavina) January 31, 2016
1/2 My thoughts are with Terry Wogan's family. Britain has lost a huge talent - someone millions came to feel was their own special friend.— David Cameron (@David_Cameron) January 31, 2016
2/2 I grew up listening to him on the radio and watching him on tv. His charm and wit always made me smile.— David Cameron (@David_Cameron) January 31, 2016
He was THE BEST...A true gentleman and so very funny . Feel very sad . RIP Sir Terry x— caroline flack (@carolineflack1) January 31, 2016