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.
Mr Obama, meanwhile, had called Mr Trump “uniquely unqualified”.
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.”
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.”
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.
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”.
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.
By Anthony Zurcher, BBC North America reporter
Donald Trump launched himself onto the national political stage by questioning Barack Obama’s US citizenship and eligibility to be president.
Mr Obama spent the last few months of the 2016 presidential election assailing Mr Trump’s fitness to occupy the nation’s highest office.
Now they’ve met for the first time ever – as president and president-elect. Talk about awkward.
Although both candidates played nice in their Oval Office get-together, there’s no denying that the two men present a clear contrast in manner and style.
In the past, the American public has displayed a tendency to seek qualities in their next president that the current occupant of the Oval Office is perceived to lack. Contrast George W Bush’s board-room manner with Bill Clinton’s improvisational chaos or Jimmy Carter’s squeaky clean reputation after Richard Nixon’s Machiavellian manoeuvres.
In this case, they have opted for Mr Trump’s blunt talk and bombast after Mr Obama’s detached professorial coolness.
Although it wasn’t on display this morning in the White House, Inauguration Day could provide a severe case of presidential personality whiplash.
With the Republicans now holding a majority in both chambers of the US Congress, Mr Trump can more easily target key Obama initiatives like such as his healthcare reforms.
Mr Trump is already setting up bilaterals ahead of his January inauguration as the nation’s 45th US president.
The office of British Prime Minister Theresa May said on Thursday that the president-elect had invited her in a phone call to visit him “as soon as possible”.
Ms May’s office said in a statement that Mr Trump had praised the trans-Atlantic “special relationship” while discussing his “close and personal connections” with the UK.
Mr Trump has family and business ties to Scotland.
The president-elect’s transition team for the 10-week period until inauguration will be led by Chris Christie, Governor of New Jersey.
Mr Trump, who has never held elected office, has said his immediate priorities will be restoring the country’s infrastructure and doubling its economic growth.
How Donald Trump won the presidency
As president-elect, he is entitled to get the same daily intelligence briefing as President Obama, which includes information on covert US operations and other data gathered by America’s 17 intelligence agencies.
Mr Trump’s team is understood to be focused on quickly filling key national security posts.
It is not yet clear who will sit his cabinet or fill senior posts in his administration, such as chief of staff, but several figures in his inner circle have been mentioned.