Long queues have formed outside polling stations, as Liberians vote in their second election since the end of a 14-year civil war.
President Ellen Johnson Sirleaf, 72, is expected to face her strongest challenge from former UN envoy Winston Tubman, a nephew of an ex-president.
Africa's first elected female head of state was awarded the Nobel Peace Prize last week.
But Mr Tubman, 70, says she has not done enough to tackle corruption.
Mrs Sirleaf won the 2005 elections after the conflict ended in 2003.
She defeated former football star George Weah, who is Mr Tubman's running mate this time.
The BBC's Jonathan Paye-Layleh in the capital, Monrovia, says hundreds of people started queuing to vote before dawn despite the heavy rain at the polling station where he is in the city centre.John Plato, 60, waiting for four hours before voting at a school in Monrovia.
"I feel very happy having cast my ballot. These elections are crucial in the eyes of Liberians and the international community and so it was important for me to vote to be part of the history-making process of the country, " he told the AFP news agency.
"All the voting places have been opened at 08:00 [local time and GMT], there are no reports of delays yet and no reports of incidents but the day is still long so we are waiting to hear from our stations in other parts of the country," National Elections Commission spokesman Bobby Livingstone told AFP.
This is first election the NEC has organised as the previous one was run by the UN.
Mrs Sirleaf had said she would only seek a single term but explained her U-turn by saying she wanted to finish the work she had started.Our correspondent says central Monrovia has been transformed since the end of the war, with roads paved and many new buildings.
While Mrs Sirleaf is well regarded by the international community, some analysts she is less popular at home and predict a tight race, possibly going to a run-off.
Her 15 challengers accuse her of not doing enough to improve the lives of ordinary people, who remain among the poorest in the world.
"One out of every three Liberians cannot feed themselves. They live in abject poverty. And they couldn't care less about the Nobel prize," said 60-year-old opposition candidate Charles Brumskine.
Mrs Sirleaf has also been criticised for backing former President Charles Taylor - currently on trial at The Hague for alleged war crimes - when he began his rebellion in 1989.
She has apologised and the pair later fell out but Liberia's Truth and Reconciliation Commission said she should be barred from holding public office.
The campaigning was generally peaceful but some 8,000 UN peacekeepers have been deployed across the country to prevent any violence.
Polling is due to end at 18:00 GMT.
Voters are also choosing members of the House of Senate and House of Representatives.
Liberia is Africa's oldest republic - it was founded in 1847 by freed US slaves, hence its name.
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