Former military ruler Muhammadu Buhari has become the first opposition candidate to win a presidential election in Nigeria.
Incumbent Goodluck Jonathan telephoned Gen Buhari, 72, on Tuesday night to congratulate him and concede defeat.
Unofficial voting tallies put Gen Buhari more than two million votes ahead of his rival.
Observers have generally praised the election, though there have been allegations of fraud.
Election results as they happened
"I promised the country free and fair elections. I have kept my word," Mr Jonathan said in a statement.
He said he had conveyed his "best wishes" to Mr Buhari, and urged "those who may feel aggrieved to follow due process... in seeking redress".
A spokesman for Gen Buhari's All Progressives Congress (APC) party praised Mr Jonathan, saying: "He will remain a hero for this move. The tension will go down dramatically."
Gen Buhari's supporters took to the streets in APC strongholds, including the northern cities of Kano and Kaduna, to sing and dance in celebration.
The APC issued a statement after the result was announced, calling for "calm, sober celebrations" and warning supporters not to attack opponents.
"He or she is not with me, whoever does that," the new president said.
The former military ruler managed to win more than 25% of votes in 24 states, meaning he avoided the possibility of a run-off with Mr Jonathan.
He dominated the country's north-western states, which have suffered most from attacks by Islamist militant group Boko Haram.
In Borno state, one of the worst-affected by Islamist violence, Gen Buhari won
It is the fourth time that Gen Buhari, 72, has sought the presidency.
He ruled Nigeria from January 1984 until August 1985, taking charge after a military coup in December 1983.
Mr Jonathan had led Nigeria since 2010, initially as acting leader before winning elections in 2011.
Nigeria has suffered from several attacks by the Islamist militant group Boko Haram, which has killed thousands of people in its drive to establish an Islamic state.
Many voters have said that they believe Gen Buhari is better positioned to defeat Boko Haram.
The verdict on Mr Buhari's 20 months as military ruler is mixed.
The European Union's top diplomat, Federica Mogherini, congratulated Gen Buhari on his victory, saying she "looked forward to working with" him.
