Earlier on Saturday, unknown attackers had attempted to blow up the national electoral headquarters in the capital Abuja but the burning petrol tanker they used stopped short of its target and failed to explode.
Saturday's violence came after the office of the ruling party's vice-presidential candidate was stormed in what police said was an assassination attempt late on Friday.
Seven Nigerian police officers on election duty were shot dead on the eve of the poll in central Nassarawa State
Witnesses said voting started late in several places, including the commercial capital Lagos, because of problems in delivering the ballots.
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"Because of the logistic problem, we have given authority to resident electoral commissioners to commence voting later than 10 o'clock as the situation demands," Maurice Iwu, the electoral commissioner, said.
In Kaduna state, European Union observers reported that voting had not started by early afternoon.
"No polling station has yet opened. No voting papers have been distributed from the [electoral commission] centre to the voting stations", Max van den Berg, head of the EU mission, said shortly before 2pm (1300 GMT).
The elections, Nigeria's first-ever transfer of power between two elected leaders, are intended to strengthen civilian rule in Africa's most populous nation.
More than 50 people have died in violence since chaotic state elections on April 14 and many more were reported dead in political violence before that vote.
The attacks underlined concerns that the elections will be undermined by violence and fraud as officials made a last-minute scramble to get ballots in place.
Speaking after Saturday's attack on the Independent National Electoral Commission offices, Sunday Ehindero, the national police chief, said: "A tanker fully loaded with substance yet to be fully identified attempted to burn down INEC headquarters in Maitama, Abuja. The tanker was unmanned."
The driver of the burning vehicle had jumped clear, leaving the throttle jammed down with a rock, but it stopped short of its target. He said the fire was extinguished before a gas cylinder intended to act as a detonator could explode.
Ehindero called on Nigerians to go ahead and vote and deplored what he called the "criminal desperation" of the unknown attackers to sabotage the election.
Opposition parties have accused the ruling People's Democratic Party (PDP) of removing ballots from secure compounds operated by the electoral commission and marking them up illegally.
"What has happened right now across the country has shown the PDP, the government, INEC and some of the law enforcement agencies are not prepared to have a free and fair election," Muhammadu Buhari said.
|Voting began late in many areas because of |
problems in delivering ballots [Reuters]
Olusegun Obasanjo, the outgoing president, on Saturday pledged that the government would not tamper with the results.
"I want to assure Nigerians that this government is a law abiding government. It has no reason to tamper with the results of elections," he said after casting his vote.
"This government and my PDP party has no reason for that. There is no party that has campaigned as much as my party."