The 80-year-old \"passed away peacefully on Saturday after a short illness\", the foundation named after him said.
His home country, Ghana, has declared a week of national mourning. Annan served two terms as UN chief from 1997 to 2006, and was awarded a Nobel Peace Prize for his humanitarian work.
He later served as the UN special envoy for Syria, leading efforts to find a solution to the conflict.
In a statement announcing his death, the Kofi Annan Foundation described him as a \"global statesman and deeply committed internationalist who fought throughout his life for a fairer and more peaceful world\".
\"Wherever there was suffering or need, he reached out and touched many people with his deep compassion and empathy.\"
Remembering the world\'s top diplomat
Obituary: Kofi Annan
Annan speaks to BBC on 80th birthday (UK only)
The career diplomat died in hospital in the Swiss city of Bern. He had been living near Geneva for several years.
He was awarded the Nobel Peace Prize in 2001 for helping to revitalise the international body, during a period that coincided with the Iraq War and the HIV/Aids pandemic.