US President-elect Donald Trump has said it was a "great honour" to meet President Barack Obama for transition talks at the White House.
Mr Obama said he was "encouraged" by their "excellent" and "wide-ranging" conversation, lasting over an hour.
During the election campaign, Mr Trump vowed to dismantle Mr Obama's legacy and he has previously questioned his US citizenship.
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But following Mr Trump's shock defeat of Hillary Clinton in Tuesday's election, Mr Obama appealed for national unity and said he was "rooting" for him.
After Thursday's behind-closed-doors meeting in the White House, Mr Obama said: "My number one priority in the coming two months is to try to facilitate a transition that ensures our president-elect is successful."
AP
He said they had discussed domestic and foreign policy and he had been "very encouraged" by Mr Trump's interest in working with President Obama's team on issues facing the US.
Mr Trump said he would "very much look forward" to dealing with President Obama in future.
"I have great respect, the meeting lasted for almost an hour and a half, and it could've, as far as I'm concerned, it could've gone on for a lot longer," the president-elect said.
"We discussed a lot of different situations - some wonderful and some difficulties."
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White House spokesman Josh Earnest said the two men did not resolve their differences but "the meeting might have been at least a little less awkward than some might have expected".
"President Obama came away from the meeting with renewed confidence in the commitment of the president-elect to engage in an effective, smooth transition," he said.
Mr Trump flew from New York on his private jet and landed at Reagan National Airport, just outside the nation's capital.
The president-elect was accompanied by his wife, Melania, who had a meeting with First Lady Michelle Obama.
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He, along with Vice-President-elect Mike Pence, then met Republican House Speaker Paul Ryan, saying they "can't get started fast enough, whether it's healthcare or immigration".
Mr Ryan described it as a "fantastic, productive meeting".
President Obama congratulated his successor in a phone call in the early hours of Wednesday.
The defeated Mrs Clinton also told supporters Mr Trump had to be given a "chance to lead".
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Despite appeals for unity, thousands took to the streets of major US cities on Wednesday. Many chanted: "Not my president."
Sixty-five people were arrested in New York, while shop windows were smashed and missiles hurled at riot police during demonstrations in Oakland, California.
In Chicago, crowds blocked the entrance to Trump Tower, chanting: "No Trump, No KKK, No Fascists USA", and a mass anti-Trump rally shut down the key 101 freeway in Los Angeles.
Mr Obama and Mr Trump have a history of mutual hostility.
Mr Trump led the charge in challenging the legitimacy of Mr Obama's presidency through the "birther" movement, which falsely claimed the Hawaii-born commander-in-chief was actually born outside the US.
The businessman also called Mr Obama "the worst president in the history of the United States".
For his part, the president famously skewered Mr Trump in person at the 2011 White House Correspondents' Association Dinner, which some have suggested may have spurred the New York billionaire to seek revenge.
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