The service is being held in front of a vociferous crowd in the FNB stadium in Johannesburg.
US President Barack Obama said Mr Mandela was a “giant of history”, describing him as the last great liberator of the 20th Century.
The former South African president died last Thursday, aged 95.
The country is observing a series of commemorations leading up to the funeral on Sunday.
The memorial service is one of the biggest gatherings of international dignitaries in recent years.
There had been fears people would be turned away, but the heavy rain left areas of the 95,000 capacity stadium empty.
Introducing the proceedings, the master of ceremonies, Cyril Ramaphosa, said that Mr Mandela’s “long walk is over… and he can finally rest”.
Mr Obama delivered his address, carried on the White House web site, to huge cheers. He said: “It is hard to eulogise any man… how much harder to do so for a giant of history, who moved a nation towards justice.”
He said Nelson Mandela had taught the world the power of action and the power of ideas, and that it had taken a man like Mr Mandela to free not only the prisoner but also the jailer.
Mr Obama said: “We will never see the likes of Nelson Mandela again. While I will always fall short of Madiba (Mr Mandela’s clan name), he makes me want to be a better man.”
On his way to the podium, President Obama shook hands with Cuban President Raul Castro, an unprecedented gesture between the leaders of two nations that have been at loggerheads for more than half a century.
In his address, Mr Castro paid tribute to Mr Mandela as the “ultimate symbol of dignity and the revolutionary struggle”.
Under his brother, Fidel, Cuba was a staunch critic of apartheid, and Mr Mandela had expressed gratitude for that support.
In his speech, UN Secretary General Ban Ki-moon said there was “sorrow for a mighty loss and celebration of a mighty life”.
He said: “South Africa has lost a hero, it has lost a father… He was one of our greatest teachers. He taught by example. He sacrificed so much and was willing to give up all he had for freedom and democracy.”