He “passed away peacefully on Saturday after a short illness”, the foundation named after him said on Saturday.
Annan served two terms as UN chief from 1997 to 2006, and was awarded a Nobel Peace Prize for humanitarian work for his efforts.
He later served as the UN special envoy for Syria, leading efforts to find a peaceful 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. He selflessly placed others first, radiating genuine kindness, warmth and brilliance in all he did.”
The diplomat, who was originally from Ghana, had been living in Geneva for several years before his death.
He was awarded the Nobel Peace Price in 2001 for helping to revitalise the international body, during a period that coincided with the Iraq War and the HIV/Aids pandemic.
Kofi Annan described his greatest achievement as the Millennium Development Goals which – for the first time – set global targets on issues such as poverty and child mortality.
However, Mr Annan was not immune to criticism. His critics blamed him for the UN’s failure to halt the genocide in Rwanda in the 1990s when he was head of the organisation’s peacekeeping operations.
Later, after the US-led invasion of Iraq, he and his son were accused of being involved in the “oil for food scandal” that led some to call for his resignation, though he was later exonerated.
In an interview with the BBC’s HardTalk to mark his 80th birthday in April, Mr Annan acknowledged the UN’s shortcomings, saying it “can be improved, it is not perfect but if it didn’t exist you would have to create it”.
“I am a stubborn optimist, I was born an optimist and will remain an optimist,” he added.
Current UN chief Antonio Guterres has been leading the tributes to his predecessor, describing Mr Annan as “a guiding force for good”.
“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.