Due to logistical problems and some cases of violence, polls have been put back to a later date in other areas.
President Muhammadu Buhari, 76, is seeking a second term. His main challenger is former Vice-President Atiku Abubakar, 72.
The election had been due last week but was delayed at the last minute.
Whoever wins in Africa’s most populous nation and largest economy will have to address power shortages, corruption, security threats, and an economic slowdown.
President Buhari cast his ballot in his hometown of Daura in the northern state of Katsina. Asked if he would congratulate his rival if he lost, he said: “I will congratulate myself.”
Most of the country was calm but there were reports of attacks by the Boko Haram Islamist militant group in the north, and voter intimidation and attempts to steal ballot boxes from some polling stations, especially in the southern states of Rivers, Lagos and Anambra.
A coalition of civil society groups reports that a total of 16 people were killed around the country – this is less than in previous elections.
Two people were arrested in the Surulere district of the commercial capital, Lagos, after voters were attacked by a group of young men “brandishing weapons: cutlasses, axes and stakes”, one witness told the BBC.
“What they were saying was that if you were not [voting for the ruling party] APC, you’ll be attacked,” Ralph Onodike, who sustained an arm injury, told the BBC.
The Independent National Electoral Commission (Inec) said that voting had not been possible in about 8,500 of the 120,000 polling stations around the country, reports the AFP news agency.
These ballot boxes were destroyed in the Isolo district of Lagos
But Inec official Festus Okoye said the commission was “generally satisfied with the process and the procedures for the conduct of these present elections”. He said 68% of polling units had opened by 10:00, according to Reuters news agency.
The initial vote was rescheduled in a dramatic press conference in the early hours of Saturday 16 February, just five hours before polls were due to have opened.
Voters were also choosing members of the House of Representatives and Senate.
The candidate with the most votes is declared the winner in the first round, as long as that person gains at least 25% of the votes in two-thirds of Nigeria’s 36 states.
There are 73 registered candidates in the presidential election, but campaigning has been dominated by the two political giants and the established party machines behind them.
The president’s All Progressives Congress (APC) has promised to take the country to the “next level”, arguing that in his first four-year term Mr Buhari has done a lot of “foundational work” that may not be immediately obvious.
Mr Abubakar and his People’s Democratic Party have pledged “to get Nigeria working again”, saying that the president has wasted the last four years.