Sir Terry had a career on television and radio spanning 50 years, including presenting Wake up to Wogan on BBC Radio 2 and the Wogan chat show.
In a statement, his family 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."
The leader of Catholics in England and Wales, Cardinal Vincent Nichols has also been responding to the news. He wrote on Twitter that he was joining millions in mourning.
Along with millions today I mourn the loss of Terry Wogan. I pray the Lord will receive him into his heavenly home there to be at peace.
— Cardinal Nichols (@CardinalNichols) January 31, 2016
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."
My tribute to Sir Terry Wogan. @BBCBreaking @sharonlhanley @BBCNews @BBCRadio2 pic.twitter.com/QLUlb7xJXM
— Jeremy Vine (@theJeremyVine) January 31, 2016
Tony Blackburn added: "I can hardly believe my old friend Sir Terry Wogan has died. RIP Terry and thanks for being a friend."
Radio 2 DJ Chris Evans said: "We are all so terribly sad upon hearing of the passing of Terry. I can't put into words how the whole Radio 2 family is feeling."
Wogan was born in Limerick and first headed into the world of banking after leaving college in 1956 but, after answering an advertisement, joined RTE where he worked as a newsreader and announcer.
He was one of the founders of Children in Need and hosted the telethon for more than 20 years, helping to raise more than £400 million for charity.
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."
In 1997 he was awarded an honorary OBE and was knighted in 2005.
He was married with two sons and a daughter.