He is said to have died "peacefully" in the early hours of this morning.
The seventh Secretary-General, who rose through the ranks of the UN, suffered with a short illness before he died.
His wife, Nane, and their children Ama, Kojo and Nina were by his side during his final days.
The Ghanaian diplomat – who won the 2001 Nobel Prize for Peace with the UN – served from January 1997 to December 2006.
A UN Twitter account posted this morning: "Today we mourn the loss of a great man, a leader, and a visionary: former UN Secretary General Kofi Annan.
"A life well live. A life worth celebrating."
A former pupil of the same secondary school as Mr Annan, who is also married to one of his grandchildren, said the family were very shocked at his sudden death.
He told the Sun Online the 80-year-old had seemed well up until he fell ill in Zimbabwe a few weeks ago.
The former Secretary-General was then taken to South Africa for treatment, before being flown to Switzerland where he then died.
His grandson-in-law added: "The last time I saw him was at the former President of Ghana's funeral.
"As he didn't live in Ghana anymore, when he would come there he would always visit his older sister, they were very close."
He was the first black African to take up the role of UN chief, and later served as the special envoy for Syria in an effort to find peace in the conflict.
Mr Annan eventually stepped down from the UN after becoming frustrated with a lack of progress in conflict resolution.
His family said in a statement this morning: "Wherever there was suffering or need, he reached out and touched many people with his deep compassion and empathy.
"He selflessly placed others first, radiating genuine kindness, warmth and brilliance in all he did.
"He will be greatly missed by so many around the world, as well as his staff at the Foundation and his many former colleagues in the United Nations systems.
"He will remain in our hearts forever."
He was the first Secretary-General to be appointed from within the organisation’s ranks.
He first joined the UN in 1962 as an administrative and budget officer with the World Health Organisation in Geneva
The popular chief was outspoken on the Iraq war, telling the BBC in 2004 the war was illegal.
Mr Annan had called on the US and UK not to invade without the support of the UN.
He added: "I have indicated it was not in conformity with the UN charter. From our point of view and from the charter point of view it was illegal."
Mr Annan served as Special Representative of the Secretary-General to the former Yugoslavia (1995-1996), and facilitated the repatriation from Iraq of more than 900 international staff and other non-Iraqi nationals (1990).
His family described him as a "son of Ghana" and said he always felt a special responsibility towards Africa.
After he stepped down from the UN he continued to work for peace as chairman of a foundation set up in his name, and as chair of The Elders, set up Nelson Mandela.
Among a great list of honours and achievements, the former UN General-Secretary was given 31 honorary degrees from institutions around the globe.
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