The first results are expected later from Nigeria’s tightest election since the end of military rule in 1999.
However, following widespread delays and some attacks on polling stations on Saturday, voting has been postponed until Sunday in parts of the country.
In other areas, voting continued through the night.
Turnout appears to be high, especially among young people who make up about a third of the 87 million eligible voters.
The election has seen an unprecedented challenge to the two-party system that has dominated Nigeria for 24 years.
Peter Obi from the relatively unknown Labour Party, Bola 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.
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.
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 at least five states, voting in some places did not begin until around 18:00 local time – three-and-a-half hours after polls were due to close.
One woman on Twitter said she used her car headlights to light up the process of voting and counting.
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 already being posted outside individual polling stations.
These results from tens of thousands of polling stations around the country will then be 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, with the first announcements expected later on Sunday.
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.
Mr Yakubu also said that armed men had attacked some polling units in the southern state of Delta and the northern state of Katsina, where voter-card verification machines were carted away.
They were subsequently replaced and security boosted to allow voting to take place, he added.
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.
The lead-up to the polls was overshadowed by a cash shortage caused by a botched attempt to redesign the currency, leading to widespread chaos at banks and cash machines as desperate people sought access to their money.
The new notes were introduced in order to tackle inflation, and also vote-buying. On the eve of the election a member of the House of Representatives was arrested with almost $500,000 (£419,000) in cash, and a list of people he was supposed to give it to, police say.
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, with another vote for state governors in March.