Turnout appears to be high, especially among young people who make up about a third of the 87 million eligible voters.
This makes it the biggest democratic exercise in Africa.
The election has seen an unprecedented challenge to the two-party system that has dominated Nigeria for 24 years.
Peter Obi from the previously little known Labour Party, Mr Tinubu from the ruling All Progressives Congress (APC) and Atiku Abubakar of the main opposition Peoples Democratic Party (PDP) are all seen as potential winners. There are 15 other presidential candidates.
A candidate needs to have the most votes and 25% of ballots cast in two-thirds of Nigeria’s 36 states to be declared the winner.
Otherwise, there will be a run-off within 21 days – a first in Nigeria’s history.
Saturday’s voting was marred by long delays at polling stations, as well as scattered reports of ballot-box snatching and attacks by armed men, especially in southern areas, where Mr Obi has strong backing.
Dr Nkem Okoli was just about to vote in the Lekki district of the biggest city Lagos when masked men attacked the polling station.
“There was pandemonium. There were bottles flying everywhere,” she told the BBC. “They broke [the ballot box]. They stole the phones of the officials. Now we can’t vote.”
In some areas, voting did not begin until around 18:00 local time – three-and-a-half hours after polls were due to close.
First-time voter Susan Ekpoh told the BBC that she spent 13 hours at her polling station in the capital, Abuja, only leaving at midnight.
She said when it got dark, election officials said they needed light to see what they were doing, so she and others.
The southern Bayelsa state was among those areas where voting was delayed until Sunday – it is not clear how many parts of the country saw voting postponed.
Harrison Rosaline said she waited for five hours to vote on Saturday in Bayelsa’s capital, Yenagoa, without seeing any election officials. But she returned, with her two-week old baby, and is delighted to have finally cast her ballot.
“I was motivated because I want a better Nigeria. I want this country to be good for everybody, including my baby,” she told the BBC.
There is tension in parts of Rivers and Lagos states, where some political parties have asked their members to go to the centres where votes are being collated, to prevent them being manipulated.
There have also been complaints over the use of the recently introduced electronic voting system, with many voters accusing electoral officials of refusing to upload the results at the polling units as they are supposed to.
However, in those areas where voting went smoothly, results are being posted outside individual polling stations.
The results from tens of thousands of polling stations around the country are being added up. An official from the electoral body in each of Nigeria’s 36 states will then travel to the capital, Abuja, where the results will be announced state-by-state.
Final results are not expected before Monday at the earliest, and possibly not until Wednesday.
At a press briefing on Saturday, electoral chief Mahmood Yakubu apologised for the delays in voting.
In the north-eastern state of Borno, Mr Yakubu said that militant Islamists had opened fire on electoral officers from a mountain top in the Gwoza area, injuring a number of officials.
Whoever wins will have to deal with a crumbling economy, high youth unemployment, and widespread insecurity which saw 10,000 killed last year.
Voters also cast their ballots for 109 federal senators and 360 members of the house of representatives.
Who are the main candidates?
Mr Obi, 61, enjoys fervent support among some sections of Nigeria’s youth, especially in the largely Christian south.
Although he was in the PDP before then, he is seen as a relatively fresh face. The wealthy businessman served as governor of the south-eastern Anambra State from 2006 to 2014. His backers, known as the “OBIdients”, say he is the only candidate with integrity, but his critics argue that a vote for him is wasted because one of the two traditional parties is more likely to win.
The PDP’s Mr Abubakar, 76, is the only major candidate from the country’s mainly Muslim north. He has run for the presidency five times before – all of which he has lost. He has been dogged by accusations of corruption and cronyism, which he denies.
Most of his career has been spent in the corridors of power, having worked as a top civil servant, vice-president and a prominent businessman.
Most people consider the election a referendum on the APC, which has overseen a period of economic hardship and worsening insecurity.
Its candidate, Mr Tinubu, 70, is credited with building Lagos during his two terms as governor until 2007.
He is known as a political godfather in the south-west region, where he wields huge influence, but like Mr Abubakar, has also been dogged by allegations of corruption over the years and poor health, both of which he denies.
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