Well-wishers saw his casket being paraded into the national sports stadium surrounded by a military band.
There are many empty seats in the stadium, which has a capacity of 60,000 people.
Mugabe is regarded as a liberator but many Zimbabweans said they would shun the ceremony in protest at the repression that marked his later years.
More than a dozen current and former leaders, including South African President Cyril Ramaphosa and Kenyan President Uhuru Kenyatta, are guests at the funeral.
Presidents, including Equatorial Guinea President Theodore Obiang Nguema, gave speeches at the stadium.
Obiang, who came to power in 1979 and has outlasted Mr Mugabe’s 37-year rule, said he was “a leader without comparison on the African continent” and praised his controversial policy of seizing white-owned farms.
“The people of Zimbabwe will forever be grateful that he took land from the whites and gave it to his people,” he said.
Like Mr Obiang, many Zimbabweans are choosing to remember the former leader’s achievements, rather than the political violence and economic chaos that marked the last years of his long presidency, reports BBC Africa correspondent Andrew Harding.
But many people in Harare were expected to avoid the ceremony as soaring inflation and unemployment grip the country.
“We are happier now that he is gone. Why should I go to his funeral? I don’t have fuel,” a Harare resident told AFP. “We don’t want to hear anything about him anymore. He is the cause of our problems.”
When and where will Mugabe be buried?
The funeral follows a row between the Mugabe family and the government over his burial.
It has now been agreed that he will be buried in the National Heroes’ Acre monument in Harare, his family says.
Family spokesman and nephew Leo Mugabe says this should be in about a month, when the new shrine to Mugabe will be built at the existing Heroes’ Acre.
Earlier plans to have a burial on Sunday appear to have been cancelled.
Mugabe, who was 95, died last week while being treated in Singapore.