David Cameron was among the first to pay tribute to Sir Terry on Twitter. The Prime Minister said: "My thoughts are with Terry Wogan's family. Britain has lost a huge talent - someone millions came to feel was their own special friend.
And the Irish Prime Minister, Enda Kenny, said: "Wogan occupied a special place in British listeners' hearts and he acted in no small way as a bridge between Ireland and Britain."
Sir Terry's family said he had died on Sunday "after a short but brave battle with cancer".
"He passed away surrounded by his family," they said in a statement. "While we understand he will be missed by many, the family ask that their privacy is respected at this time."
Perhaps most famous for his breakfast radio show Wake Up Wogan, on TV he also presented the Wogan chatshow and became the sardonic voice of UK coverage of the Eurovision Song Contest.
BBC director-general Tony Hall said Sir Terry was "truly a national treasure". "He was a lovely, lovely man and our thoughts are with his wife and family," he said.
"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."