Trump, a New York billionaire, and Sanders, a Vermont senator and democratic socialist, were favored to win their races, riding a wave of voter anger at traditional politicians seeking their parties' presidential nomination.
Trump was under pressure to deliver a victory after he was beaten in the first nominating contest - the Iowa caucuses on Feb. 1 - by Texas Senator Ted Cruz despite having had a big lead in pre-caucus polls. Trump posted a video on Facebook urging his supporters to vote.
"The polls don't mean anything if you don’t get up, don’t get out, don’t vote," he said. "We have to vote. You have to make this change."
New Hampshire is the second state in the process of picking party nominees for the Nov. 8 election to replace President Barack Obama. The polls were to close at 7 p.m. EST (0000 GMT) and New Hampshire officials predicted a historic high turnout of about 550,000.
Trump's rivals were dueling for second place as the last undecided voters made up their minds.
Ohio Governor John Kasich, who held more than 100 town hall events in the state, appeared to be getting some of the late deciders but Cruz, former Florida Governor Jeb Bush, Florida Senator Marco Rubio and New Jersey Governor Chris Christie also felt good about their chances.
Kasich, who maintained a positive message in contrast to the angry rhetoric of Trump, told CNN a strong showing for him would show that "maybe we ought to spend more time trying to sell who we are rather than trashing somebody else."
On the Democratic side, Clinton fought to keep it close against Sanders.
Despite being an overwhelming front-runner, Clinton's razor-thin victory in Iowa made her look vulnerable and a double-digit loss to Sanders in New Hampshire would add pressure on her to improve her performance. There was some talk among Clinton loyalists of a possible campaign shakeup.
At a polling location at McDonough Elementary School, just outside downtown Manchester, many of the midday voters said they would be supporting Sanders.
Kristian Gustafson, 47, said he felt great about Sanders’ chances in the state and likes the candidate's position against large campaign finance donations.
"Campaign finance is a big thing for me and he’s spent more time talking about that than anyone else,” Gustafson said.
Rhoda Goley, 47, backed Clinton, saying she liked "that she’s run on a platform of children and families."
Rubio needed to follow up his strong third-place showing in Iowa with a top-tier finish in New Hampshire to bolster his argument that he is the candidate around whom the party's leadership and wealthy donors should rally.
But a shaky debate performance by Rubio on Saturday night came at a bad time. Rubio aides insisted no harm was done and that his crowds were as big as ever.
A WMUR-CNN poll on Monday showed Trump leading in New Hampshire with the support of 31 percent of those planning to vote in the Republican primary. Rubio was second at 17 percent, followed by Cruz at 14 percent and Kasich at 10 percent, with a margin of error of plus or minus 5.2 percentage points.
Bush and former tech executive Carly Fiorina trailed in the single digits.
Trump, who has courted controversy by deriding Mexican immigrants and promising to ban Muslims from entering the United States, spent the final campaign hours in New Hampshire insulting his rivals.
In an interview with MSNBC, he called Rubio "confused," Bush a "loser," Clinton "evil" and Cruz "nasty."
At a campaign event on Monday, the real estate mogul gleefully repeated an audience member's description of Cruz as a "pussy" because Cruz said he was more hesitant than Trump about supporting torturing the country's captured enemies.
"Nothing Donald says surprises anyone," Cruz said as he visited a polling station. "He didn’t like that he lost in Iowa."
At a polling station in the town of Derry on Tuesday morning, Clinton bumped into the husband of former Hewlett-Packard CEO Fiorina, who has repeatedly derided Clinton's marriage to former U.S. President Bill Clinton as loveless.
"Well, give my best to Carly," Clinton said to Frank Fiorina after they had swapped pleasantries about the marvels of democracy. "Want to get a picture?"
They grinned for cameras.
Primary votes already were counted in Dixville Notch, a town of about a dozen people that prides itself on being the first in the state to vote. Sanders won all four Democratic votes there while in the Republican race Kasich beat Trump, 3-2.
(Reporting by James Oliphant and Amanda Becker; Writing by Steve Holland and James Oliphant; Additional reporting by Jonathan Allen, Susan Heavey and Doina Chiacu; Editing by Toni Reinhold, Alistair Bell and Bill Trott)
Watch a live broadcast of Tuesday's primary results, as aired on ABC News, below.
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