He "passed away peacefully on Saturday after a short illness", the foundation named after him said on Saturday.
Annan was the first black African to take up the role of the world's top diplomat, serving from 1997 to 2006.
He later served as the UN special envoy for Syria, leading efforts to find a peaceful solution to the conflict.
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. He selflessly placed others first, radiating genuine kindness, warmth and brilliance in all he did."
Annan's tenure as UN secretary-general coincided with the Iraq War and the HIV/Aids pandemic.