Gen. Muhammadu Buhari has hailed his victory as a vote for change and proof the nation has embraced democracy.
He also praised outgoing President Goodluck Jonathan as a “worthy opponent” who peacefully relinquished power.
Muhammadu Buhari beat Goodluck Jonathan by 15.4 million votes to 13.3 million.
Observers have generally praised the election, though there have been allegations of fraud.
Muhammadu Buhari, of the All Progressives Congress (APC) party, has become the first opposition candidate to win a presidential election in Nigeria.
He said: “President Jonathan was a worthy opponent and I extend the hand of fellowship to him.
“We have proven to the world that we are people who have embraced democracy. We have put the one-party state behind us.”
The former military ruler added: “Your vote affirms that you believe Nigeria’s future can be better than what it is today. You voted for change and now change has come.
“You, Nigerians, have won. The people have shown their love for this nation and their belief in democracy.”
On March 31, Goodluck Jonathan said in a statement: “I promised the country free and fair elections. I have kept my word.
“Nobody’s ambition is worth the blood of any Nigerian. The unity, stability and progress of our dear country is more important than anything else.”
Goodluck Jonathan said he had conveyed his “best wishes” to Muhammadu Buhari, and urged “those who may feel aggrieved to follow due process… in seeking redress”.
However, analysts say the margin of victory is likely to prevent any successful legal challenge.
Goodluck Jonathan must officially hand over power on May 29.
This election was the fourth time that Muhammadu Buhari, 72, had sought the presidency.
He ruled Nigeria from January 1984 until August 1985, taking charge after a military coup in December 1983.
Goodluck Jonathan had led Nigeria since 2010, initially as acting leader before winning elections in 2011.
Nigeria has suffered from 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 Muhammadu Buhari is better positioned to defeat Boko Haram.
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