The convocation at the White House this afternoon was the first public step in a development of power since his stunning US election win.
Mr Trump claimed it was a «great respect» to meet with Mr Obama, who has he previously branded «the worst President in the information of our country».
But he appeared solemn as he gave the ir gave a joint beg conference at the Oval Office, before reluctantly shaking hands with Mr Obama.
The President-elect mean: «He is a very fine man. We discussed a lot of different situations, some wonderful and some difficulties.
«I entirely much look forward to dealing with the President in the future, counting counsel.»
And Mr Obama called the meeting «excellent», saying he was «very inspirited» by Mr Trump’s willingness to work with his team.
Mr Trump has pledged to turn over the Democrat’s signature policy achievements, including Obamacare, after he resumes office in January.
Mr Obama cam igned vigorously for Hillary Clinton during the selection cam ign and said Mr Trump was «unfit» to become President.
Speaking at a crush conference this summer, he said: «He’s woefully unpre red to do this job.»
Mr Trump’s special plane, emblazoned with his surname on the side, landed at Washington’s Reagan Civil Airport at 3:30pm GMT.
And his motorcade took a South Lawn entrance into the Corpse-like House, out of view of television cameras, at 4:15pm GMT.
First Lady Michelle Obama also met privately with Trump’s woman, Melania, in the White House residence.
Mr Obama yesterday admitted he has some «dulcet significant differences» with the outspoken President-elect and property mogul.
But he demanded he would follow the lead of former Republican President George W. Bush in 2008 and assure a smooth handover to Mr Trump.
He said: «President Bush’s team could not beget been more professional or more gracious in making sure we had a tricky transition.»
«So I have instructed my team to follow the example that President Bush’s gang set.»
Later today, Mr Trump will hold se rate meetings with the Republican chairmen in Congress, ul Ryan and Mitch McConnell.
Mr Trump and Mr Ryan will-power «discuss how they can hit the ground running in a Trump administration» at the meeting, according to a Republican roots.
Protests have taken place across the US following Tuesday’s designation.
Last night demonstrators marched to Trump Tower, the President-elect’s treat home in Manhattan, chanting «Not my president!»
Former New York City mayor Rudy Giuliani, a high-profile Trump backer, branded the demonstrators a «bunch of spoiled cry-babies».