Michelle Obama met privately in the White House residence with Mr Trump's wife Melania
Barack Obama has said he had an "excellent conversation" with President-elect Donald Trump at the White House, as the handover of power begins.
Mr Obama said he was encouraged by Mr Trump's willingness to work with his team on the issues facing the country, adding it was important for all "to now come together".
The president-elect said their talks, which lasted for 90 minutes, were wide-ranging and called the meeting a "great honour".
The respectful tone adopted by the two men was in contrast to the bitter and acrimonious exchanges which characterised the election campaign, which saw Mr Trump beat Hillary Clinton.
The White House said afterwards that while the pair did not resolve all their differences at the meeting, the talks "might have been at least a little less awkward than some might have expected".
Mr Trump said he looked forward to more meetings with the president before his inauguration in January and that they talked about "a lot of different situations, some wonderful and some difficulties".
He said the president explained "some of the great things that have been achieved", but did not elaborate.
Mr Obama was critical of Mr Trump throughout the campaign, saying he was unfit to serve as commander in chief.
The President-elect, meanwhile has been accused of feeding the 'birther' conspiracy theories about President Obama, as well as a string of negative comments during the campaign.
If the Republican makes good on his campaign promises he will wipe away much of what Mr Obama has done in office, including his signature healthcare reforms and the landmark nuclear agreement with Iran.
But the pair continued the conciliatory tone that has emerged since Mr Trump's shock victory, with the Republican saying he was looking forward to benefiting from Mr Obama's "counsel" in the future.
Mr Obama pledged his administration would "do everything we can to help you succeed, because if you succeed, then the country succeeds".
He said: "My number-one priority in the coming two months is to try to facilitate a transition that ensures our president-elect is successful."
The two men ended their historic encounter, which was originally scheduled to last for no more than 15 minutes, with a handshake and refused to take questions from the journalists present.
First Lady Michelle Obama also met privately in the White House residence with Mr Trump's wife Melania, while Vice President Joe Biden will meet Vice President-elect Mike Pence later on Thursday.
Spokesman Josh Earnest said Mrs Obama and Mrs Trump spoke about the challenges of raising children in the White House. Mr Trump's 10-year-old son Barron will become a teenager during his father's first term.
The transition meeting comes after a night of protests sparked by the Republican's surprise victory over Mrs Clinton in one of the most divisive elections in US history.
Demonstrators in more than 10 cities across the nation chanted "Not my president", burned a papier mache head of the president-elect, hit a pinata with his likeness and carried signs that said "Impeach Trump".
A second round of protests is planned on Thursday evening.
As well as the talks in the Oval Office, Mr Trump met House Speaker Paul Ryan and Senate Majority Leader Mitch McConnell to discuss the Republicans' legislative agenda.
Speaking afterwards he said he would do "spectacular" things for the American people and would work quickly on issues like healthcare and immigration.
The Republicans now control both the Senate and the House of Representatives, meaning Mr Trump will find it easier to get things done once he is sworn in.
But Democratic senator Elizabeth Warren has said the party would fight him on social issues and financial regulation.
As he prepares to move to the White House in January, Donald Trump has also contacted the leaders of the UK, South Korea, Japan, Egypt, Australia, India, Turkey, Israel and Mexico.
He also invited Taoiseach Enda Kenny to attend the St Patrick's Day celebrations at the White House, as is the tradition.
Mr Kenny is said to have Mr Trump had a 10-minute conversation with the Taoiseach on Wednesday night.
In a statement, a spokesperson said: "The President Elect confirmed to the Taoiseach that in the spirit of the strong ties between the two countries, the long standing tradition of Taoisigh attending the White House for St Patrick's Day celebrations would continue and extended an invitation to the Taoiseach in that regard for next year 2017."