Billionaire Mr Trump is likely to get more than twice the number of votes of the next Republican candidate.
Senator Bernie Sanders, who beat Democratic rival Hillary Clinton by a huge margin, said his victory showed people wanted "real change".
Both candidates are riding on a wave of discontent with mainstream politics.
The aim of the primary race is to choose which candidates will represent the Republican and Democratic parties in November's presidential election.
On Tuesday Ohio Governor John Kasich came second in the Republican vote, with former Florida Governor Jeb Bush, Texas Senator Ted Cruz and Florida Senator Marco Rubio all vying for third place.
New Hampshire is the second state to choose delegates in the long nomination battle following last week's Iowa caucuses, which were won by Mr Cruz for the Republicans and Mrs Clinton for the Democrats.
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