It is said to be the most closely fought election since independence.
The election was delayed by six weeks to allow the army to recapture territory from militant Islamist group Boko Haram.
The two main presidential candidates have pledged to prevent violence during the election and its aftermath.
The People’s Democratic Party (PDP) has dominated Nigerian politics since 1999, but the All Progressives Congress (APC) is viewed as a serious challenge.
Some 800 people were killed after the 2011 contest between Goodluck Jonathan and Gen. Muhammadu Buhari, a former military ruler.
The polls opened at 08:00 AM local time.
Voters in 36 states and the Federal Capital Territory of Abuja – the capital – will also elect members of the House of Representatives and the Senate.
On March 27, the Nigerian army said it had retaken the town of Gwoza, believed to be the headquarters of Boko Haram, one of the last places still under its control.
UN Secretary General Ban Ki-moon said on Friday: “The international community has high expectations that Nigeria will provide leadership in setting a high standard for this election.”
He called on Nigerians – in Africa’s most populous nation – to vote in large numbers.
Ban Ki-moon added that he hoped the presidential and parliamentary elections would be “transparent, inclusive and peaceful”.
Campaign group Human Rights Watch says Boko Haram has killed some 1,000 people this year alone.
On March 25, army chief Kenneth Minimah said adequate security arrangements had been made for the polls.
On March 26, the government closed its land and sea borders for the election.
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