Last Updated Feb 9, 2016 5:03 PM EST
5:02 p.m. Kasich will continue onward: Addressing concerns about his ability to mount a campaign after New Hampshire, Kasich told reporters, "We're going to be just fine. We have the money and the organization just like we always do. Sometimes, we put it together with bailing wire and duct tape, but we always seem to get it done."
12:20 p.m. Ben Carson is heading to South Carolina: CBS News' Erica Brown confirmed with a source close to Carson's campaign that the retired neurosurgeon is skipping a New Hampshire primary event today and will instead head to South Carolina, which holds its primary on Feb. 20.
10:43 a.m.Trump to announce foreign policy advisers soon: On Fox News' "America's Newsroom," Trump said he plans to release his list of foreign policy advisers in about two weeks. That was his response after he was asked to name specific people he speaks to about international issues.
"I think I know more about foreign policy than anybody running," he said.
10:35 a.m.Kasich celebrates Dixville Notch win: Kasich stopped at a crowded Red Arrow diner in Manchester with his wife and got behind the counter, served breakfast and poured coffee. The Ohio governor appeared relax and positive, refused to attack his opponents and joked that he celebrated his win in Dixville Notch with his wife this morning by drinking champagne.
Kasich said that the next president should be chosen by coal miners, not billionaires.
10:00 a.m.Christie and a second-place finish? Christie just stopped by the Webster Elementary School polling site in Manchester to greet volunteers and voters outside. Asked by reporters what a second-place finish would mean, he responded, "A second-place finish would do a lot for me here. I mean, second-place finishes make presidents, as we've seen with Bill Clinton, as we saw with Barack Obama."
9:30 a.m. Primary Day:All nine Republican candidates and the two candidates in the Democratic race are out on the trail today, holding at least 35 events. Many of them are making multiple stops at polling sites to greet Granite State voters.
The votes started to flow in slowly at midnight when three villages cast the first primary ballots. Ohio Gov. John Kasich, for example, won in Dixville Notch with the support of nine voters at midnight.
"We're going to work hard until all the polls close to continue our success throughout New Hampshire and then take the campaign to South Carolina and beyond," Kasich said in a statement.
Former Florida Gov. Jeb Bush has a similar strategy.
Closing my time in New Hampshire the same way I began: giving it my all, speaking to every voter and being true to what I believe.— Jeb Bush (@JebBush) February 9, 2016
Bush stopped by a high school in Bedford before 9 a.m. and shook hands and exchanged hugs with supporters. Bush told CBS News' Major Garrett he felt good about his prospects tonight, knocked Trump and said he was going to South Carolina no matter the result in New Hampshire.
Donald Trump, who has consistently led the GOP pack in New Hampshire, spent his morning tweeting and appearing on Fox News' "Fox and Friends."
I know the "Governors" and Jeb Bush, who has gone nasty with lies, is by far the weakest of the lot. His family used private eminent domain!— Donald J. Trump (@realDonaldTrump) February 9, 2016
He said he has the best chance of winning the general election.
"States that are never in play for Repubs will be won by me. Great!" he tweeted.
Outside of a polling site at an elementary school in Manchester, Hillary Clinton greeted voters and took pictures with them. Asked how she feels about a comeback, she said, "This is a great process, and as I've said over the last couple days, we're going to keep working literally until the last vote is cast and counted."
Sen. Bernie Sanders, I-Vermont, won the support of two out of three of the towns that voted at midnight.
CBS News' Steve Chaggaris, Arden Farhi, Katiana Krawchenko, Erica Brown, Sopan Deb, Jacqueline Alemany and Catherine Cannon contributed.