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6 year oldCurrent UN chief Antonio Guterres hailed him as "a guiding force for good" and Russian President Vladimir Putin described a "remarkable person".
Ex-US President Barack Obama said Annan had always pursued "a better world".
The Ghanaian national served as UN chief from 1997 to 2006 and is the only black African ever to hold the post.
Since then he has served as the UN special envoy for Syria, leading efforts to find a solution to the conflict.
The career diplomat died in hospital in the Swiss city of Bern. He had been living near Geneva for several years.
He "passed away peacefully on Saturday after a short illness", the Kofi Annan Foundation said.
He was a "deeply committed internationalist who fought throughout his life for a fairer and more peaceful world", the statement added.
Mr Guterres led the tributes to his predecessor. "In many ways, Kofi Annan was the United Nations. He rose through the ranks to lead the organisation into the new millennium with matchless dignity and determination," he said in a statement.
UN high commissioner for human rights Zeid Ra'ad Al Hussein said in a tweet he was grief-stricken over Annan's death:
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