Billionaire businessman Donald Trump topped a crowded Republican field in New Hampshire primaries Tuesday, while Senator Bernie Sanders trounced Hillary Rodham Clinton in the closely watched first US presidential primary election.

Trump led with 35 per cent of the vote with more than 70 per cent of ballots counted. On the Democratic side, Sanders took 60 per cent of the vote to 39 per cent for Clinton.

Sanders said his victory serves notice to the political and economic establishment in the United States.

"We must tell the billionaire class and the 1 per cent that they cannot have it all," said Sanders, who serves in the US Senate as a Vermont independent.

"The people of New Hampshire have sent a profound message to the political establishment, to the economic establishment and, by the way, to the media establishment."

Sanders, a self-described democratic socialist, said the American people will not continue to accept a corrupt campaign finance system and rigged economy in which ordinary Americans work longer hours for less money.

Meanwhile, Clinton, who had long been considered the Democratic Party's heir apparent, vowed to continue her campaign and painted herself as the pragmatic choice who had a chance to enact her policies.

"I will work harder than anyone to actually make the changes that make your lives better," Clinton said.

The Clinton campaign expressed optimism moving forward to the next contests in Nevada and South Carolina and noting that many more of the delegates needed to secure the nomination would be awarded next month in states that demographically better reflect the US and the Democratic Party.

"We're going to fight for every vote in every state," she said. "We're going to fight for real solutions that make a difference in people's lives."

Trump celebrated his victory by repeating promises to renegotiate trade deals, build a wall along the US-Mexican border and deport illegal immigrants.

"The world is going to respect us again, believe me," Trump said as his lead stretched to more than 19 percentage points over his nearest competitor.

"We are going to start winning again. You are going to be so happy," he said. "We are going to make America great again, maybe greater than ever before."

Ohio Governor John Kasich, a moderate Republican, came in second after devoting nearly all of his campaign resources to the north-eastern state.

"We are going to solve the problems in America not by being extreme," he said, but by "reminding everybody that we are dedicated to shining up America."

The other Republican candidates were locked in a close contest for third place in what is considered a make-or-break moment for many of them. The result could prolong the process of winnowing the field, with contests in South Carolina and Nevada later this month and in more than a dozen states on March 1.

Texas Senator Ted Cruz, who finished first in the Iowa caucuses that marked the first in the series of state-by-state party votes, had 12 per cent of the vote, while former Florida governor Jeb Bush was at 11 per cent. They were followed closely by Florida Senator Marco Rubio at 10 per cent.

Rubio, who had been seen as poised to breakout as the party establishment choice, blamed himself for his poor performance after exceeding expectations with a third place finish in Iowa.

"Our disappointment is not on you, it is on me," he told supporters, pointing to mistakes in a debate on Saturday. "That will never happen again."

Meanwhile New Jersey Governor Chris Christie said he would return home to re-evaluate his campaign, a sign that he could soon drop out of the race.

The vote in New Hampshire, which has a population of 1.3 million people in an area slightly larger than Israel, is the first in the 2016 presidential campaign using the state primary method, following the February 1 Iowa caucus meetings.

All registered New Hampshire voters - not just registered Democrats and Republicans - are allowed to participate in the primary. Unaffiliated voters are classified as independents and may choose to in either party.

About 43 per cent of New Hampshire voters are independents, while Democrats and Republicans make up close to 30 per cent each.

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