Donald Trump is due to meet with Barack Obama to formally begin the process of presidential transition after a victory that left many stunned, and other elated.

The sitting President spoke from the White House rose garden the afternoon after one of the most startling upsets in modern US politics, and ensured the American people of a peaceful transition of power.

"I have instructed my team to follow the example that President Bush's team set eight years ago, and work as hard as we can to make sure that this is a successful transition for the President-elect," Mr Obama said.

"The peaceful transition of power is one of the hallmarks of our democracy. And over the next few months, we are going to show that to the world."

Earlier Wednedsay, in the wake of her defeat, Hillary Clinton appeared at the Wyndham Hotel in New York to offer a formal concession speech and to urge the country to unite.

"This is not the outcome we wanted or we worked so hard for," she said, "and I’m sorry that we did not win this election for the values we share and the vision we hold for our country."

Barack Obama delivers a statement after Donald Trump was elected next U.S. president (REUTERS) (Reuters)

“Our campaign was never about one person or one election. It was about the country we love, about building a country that was hopeful, inclusive and big-hearted. I still believe in America and I always will.”

But as Ms Clinton bowed out - this surely the end of a political career that dates back four decades - the man who defeated her was quick to start preparing for the job of the presidency.

Reports said that the 70-year-old Republican, who had himself called for national unity in a rare conciliatory address to his supporters in the early hours of Wednesday morning, was meeting with his transition team in New York.

“Thank you everybody. Sorry to have kept you waiting, it’s a complicated business,” said Mr Trump, who looked as surprised as anyone.

Hillary Clinton urges supporters to give Donald Trump a chance to lead

At around 3am in New York, Mr Trump, who secured a tally of 279 electoral votes compared to 229 for Ms Clinton, said she had called him moments earlier to offer her congratulations.

“She congratulated us - it is us - on our victory, and I congratulated her and her family for a very hard fought campaign,” he said.

Mr Trump’s first job on Wednesday, according to reports, was to begin the process of selecting potential cabinet members for his government. One senior Republican official, Mitch McConnell, said that among the new president’s priorities in office would be to scrap Obamacare.

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Mr Trump is now set to receive the same intelligence briefings made available to Mr Obama.

The president, whose clashes with Mr Trump have played out over the years, during which the New York tycoon suggested the president had not been born in the US and was therefore not qualified to sit in the White House, reflected on the double-edged sword of joy and disappointment that accompanied any political campaign.

Speaking in the Rose Garden alongside Vice-President Joe Biden, Mr Obama said it was no secret that he disagreed with Mr Trump on a number of issues. Still, he vowed to set aside those differences for the greater good of the US.

"Now, everybody is sad when their side loses an election, but the day after we have to remember that we're actually all on one team," he said, speaking alongside Vice President Joe Biden in the White House Rose Garden. "We're not Democrats first. We're not Republicans first. We are Americans first. We're patriots first."

President-elect Donald Trump will have an early Capitol Hill honeymoon with Republican majorities in both chambers of Congress when he takes office in January, but a long-term romance may be more challenging.

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The Associated Press pointed out that a president whose party controls both the Senate and House of Representatives can generally count on getting things done fairly quickly and Mr Trump likely will not be an exception, despite starting with unusual handicaps.

Many Republicans in Congress only backed Mr Trump after he became the candidate. Some never fell in line. He offended and attacked others while running as a Washington outsider, including House of Representatives Speaker Paul Ryan, who did not campaign with him.

On Wednesday, party leaders were quick to back the victor, and they are expected to find more common ground than differences once he takes office.