The Ghanaian national was a career diplomat who projected quiet charisma and was widely credited for raising the world body's profile in global politics during his two terms as head of the UN from 1997 to 2006.
The first secretary general from sub-Saharan Africa, Annan led the United Nations through the divisive years of the Iraq war and was later accused of corruption in the oil-for-food scandal, one of the most trying times of his tenure.
Annan "astutely guided the United Nations organisation into the 21st century defining an ambitious agenda that had made the UN truly indispensable to peace, prosperity and human dignity around the world," Annan's successor as UN secretary general, Ban Ki-moon, said in a statement.
Annan's family said he had passed away peacefully on Saturday after a short illness.
Annan, who lived not far from the UN European headquarters in Geneva, died in a Bern hospital, Swiss media reported.
"Honestly speaking, we are in a mess," he told AFP last December, warning that "today, leaders are going in the wrong direction... leaders are withdrawing."
Despite his criticism, current and former world leaders voiced their admiration for Annan.
Russian President Vladimir Putin hailed his "wisdom and courage", while German Chancellor Angela Merkel celebrated the "exceptional statesman in the service of the global community."
His foundation told the ATS news agency Sunday his funeral would be a private event, but did not provide further details. It said a separate UN ceremony would be held at a later date.
President Donald Trump has as yet made no comment, but his Secretary of State, Mike Pompeo on Sunday hailed a life spent "advocating for peace and human dignity".
And US ambassador to the UN Nikki Haley said Annan "worked tirelessly to unite us and never stopped fighting for the dignity of every person".
Former US president Barack Obama earlier said Annan "embodied the mission of the United Nations like few others".
"Kofi Annan was a truly great UN Secretary-General," former US president Bill Clinton and former secretary of state Hillary Clinton said in a statement.