The Prime Minister said: “A great light has gone out in the world. Nelson Mandela was a towering figure in our time: a legend in life and now in death – a true global hero.
“Meeting him was one of the great honours of my life.”
US President Barack Obama said he had achieved “more than can be expected by any man”.
He said: “Today he has gone home and we have lost one of the most inspirational, courageous and profoundly good human beings that any of us will share time with on this Earth.
“He now no longer belongs to us, he belongs to the ages.”
South African president Jacob Zuma made the emotional announcement live on television, in which he said Mandela was now at peace.
He added: “Our nation has lost its greatest son.”
Irish Taoiseach Enda Kenny said a “great light has been extinguished”.
He said: “The boy from the Transkei has finished his long walk. His journey transformed not just South Africa, but humanity itself.”
UN Secretary General Ban Ki Moon said: “Only because of such a great man like Nelson Mandela is it possible that particular people in Africa and elsewhere are able to enjoy freedom and human dignity.
“We have to learn the wisdom and determinations and commitment of Mr Mandela to make this world better for all.”