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", adding: "The world thanks you for sharing Nelson Mandela with us".
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".
The first speaker, friend and fellow Robben Island inmate Andrew Mlangeni, said Mr Mandela had "created hope when there was none".
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."
Many people stood in the rain waiting for several hours to get into the stadium, the BBC's Pumza Fihlani reports from the scene.
She says the crowds are in high spirits - singing and dancing, stomping their feet - and the stadium has the feel of a political rally.
Rather than seeing the rain as a dampener, many in South Africa have welcomed it.
Defence Minister Nosiviwe Mapisa-Nqakula said: "It's a blessing from the ancestors welcoming a son of the soil."
One of those attending, Shahida Rowe from Johannesburg, told the BBC: "The core of Mandela's life was humanity. That is why I am here today and the world is celebrating.
"Thanks to him, I was recognised as a human being."
Mr Mandela's widow, Graca Machel, arrived at the stadium to huge cheers as she was shown on the big screen.
There were cheers too of "Winnie! Winnie!" for ex-wife Winnie Madikizela-Mandela, who hugged and kissed Graca Machel.
However, there were boos for current South African President Jacob Zuma.
He will make the keynote address. Brazilian President Dilma Rousseff, Chinese Vice-President Li Yuanchao, President Hifikepunye Pohamba of Namibia and Indian President Pranab Mukherjee also made speeches.
The BBC's Mike Wooldridge in Johannesburg says the line-up of world leaders is a sign of how South Africa has seen its partners and place in the world since Nelson Mandela became president.
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."
Many people stood in the rain waiting for several hours to get into the stadium, the BBC's Pumza Fihlani reports from the scene.
She says the crowds are in high spirits - singing and dancing, stomping their feet - and the stadium has the feel of a political rally.
Rather than seeing the rain as a dampener, many in South Africa have welcomed it.
Defence Minister Nosiviwe Mapisa-Nqakula said: "It's a blessing from the ancestors welcoming a son of the soil."
One of those attending, Shahida Rowe from Johannesburg, told the BBC: "The core of Mandela's life was humanity. That is why I am here today and the world is celebrating.
"Thanks to him, I was recognised as a human being."
Mr Mandela's widow, Graca Machel, arrived at the stadium to huge cheers as she was shown on the big screen.
There were cheers too of "Winnie! Winnie!" for ex-wife Winnie Madikizela-Mandela, who hugged and kissed Graca Machel.
However, there were boos for current South African President Jacob Zuma.
He will make the keynote address. Brazilian President Dilma Rousseff, Chinese Vice-President Li Yuanchao, President Hifikepunye Pohamba of Namibia and Indian President Pranab Mukherjee also made speeches.
The BBC's Mike Wooldridge in Johannesburg says the line-up of world leaders is a sign of how South Africa has seen its partners and place in the world since Nelson Mandela became president.
The memorial service, which had been due to start at 11:00 (09:00 GMT), will last about four hours, according to the official programme.
Correspondents say that the heavy rain, security and transport issues and the fact that Tuesday was not declared a national holiday have kept the numbers down.
Mr Mandela's body will lie in state in Pretoria on the following three days and a state funeral takes place on Sunday in his home village of Qunu in Eastern Cape province.
More than 100 current or former heads of state or government will attend the funeral or the national memorial, according to the South African government.
Among those not attending the memorial events will be Israel's Prime Minister Benjamin Netanyahu, who cited high travel and security costs.
However there will be suspicion that he wishes to avoid the potential for anti-Israeli protests, the BBC's Middle East Correspondent Kevin Connolly reports.
Israel had extensive links with the old apartheid regime and many South Africans identify strongly with the Palestinian cause, our correspondent adds.
Source: BBC News
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