10:59 p.m. Ted Cruz declared the "real winner" in New Hampshire the "conservative grassroots," but also congratulated Trump for his victory.

A few minutes later he went on the attack against Trump again, albeit without naming names.

"We need to be unequivocally against Obamacare. We cannot win with nominating a candidate with the same policy on Obamacare as Bernie Sanders. And for Pete's sake, we don't need more deals," he said.

Cruz said the exact results were unknown, but "Right now it appears that we are effectively tied for third in the state of New Hampshire. That was the result that all of us were told was impossible."

10:52 p.m. After days of defending his performance in Saturday's Republican debate, Rubio finally apologized in his speech following the New Hampshire primary. With 70 percent of votes counted, he is in fifth place -- far from the second-place finish his campaign had hoped for.

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"I'm disappointed with tonight. I want you to understand but I want you to understand something, I want you to understand something. Our disappointment tonight is not on you. It's on me. It's on me. I did not do well on Saturday night, so listen to this: That will never happen again," Rubio said. "Let me tell you why it will never happen again. It's not about me, it's not about this campaign. It is about this election. It is about this election. It is about what's at stake in this election."

He also admitted his campaign "did not wind up where we wanted to be," but promised to press onward and said he would return to New England in a few weeks.

10:46 p.m. It could be a few more hours before Jeb Bush knows whether he finished in third, fourth or fifth place in New Hampshire, but either way he's staying in the race.

"This campaign is not dead. We're going on to South Carolina," he promised supporters during his speech.

His communications director, Tim Miller, said the Bush team is "really confident" that Bush will do well in South Carolina, in part because the state has a closed primary system that will prevent independent voters from flocking to Trump. He also suggested the state will not prove friendly to Kasich, the second-place winner in New Hampshire.

Kasich's team does, in fact, seem focused on the March 8 Michigan primary.

10:32 p.m. Mired in sixth place, Gov. Chris Christie said he and his family will go home to New Jersey Wednesday to "make a decision on our next step forward" instead of continuing straight to South Carolina.

"We're going to take a deep breath see what the final results are tomorrow...because so many New Hampshire residents came out to vote today, it's going to take a while to count these votes so we want to see exactly what happens," Christie said. "By tomorrow morning, tomorrow afternoon we should know...that's going to allow us to make a decision about how we move from here in this race but there's no reason to go to sit in South Carolina in a hotel room to hear that."

"We will make a decision on our next step forward based upon the results," he added.

Christie also said, "We leave New Hampshire tonight without an ounce of regret, not for the time we've spent, and for the thousands of people tonight in New Hampshire who will have voted for us. We thank each and every one of them."

10:20 p.m. "Something big happened tonight and let me tell you what it is...we have had tens and tens of millions of dollars spent against us with negative advertising," Kasich said at his speech in New Hampshire. "We never went negative because we have more good to sell than to spend our time being critical of somebody else."

"And maybe, just maybe, at a time when clearly change is in the air, maybe just maybe we are turning the page on a dark part of American politics because tonight the light overcame the darkness of negative campaigning and you made it happen," he added.

Much of the rest of Kasich's speech talked about how he had been influenced by campaigning in New Hampshire (106 town halls) and the people he met. He closed with a message encouraging people to "slow down" and talk to their family and neighbors. It fit with the broader message, which he touched on in his speech, about uniting the country. He pledged to "spend my time listening and healing and helping to fix our country."

There was room for a quick crack at the Democrats, though.

"Bernie talked so long I thought he was going to hit his 77th birthday," Kasich joked.

10:09 p.m. In his victory speech, Trump suggested that he had managed to turn around his New Hampshire ground game in the week since it hurt his campaign in Iowa.

"We learned a lot about ground games in one week, I've got to tell you," Trump said, before launching into an abbreviated version of his stump speech.

On the Democratic winner, Bernie Sanders, Trump warned that he wants to "give away our country."

"Congratulations to Bernie in all fairness," he said. "We wish him a lot of luck but we are going to make America great again but we're going to do it the old fashioned way" -- by beating countries like China and Japan.

He wrapped up by saying, "Thank you new Hampshire, thank you! We are going now to South Carolina, we're going to win in south Carolina, I love you all, thank you very much."

9:54 p.m. The updated delegate count: 10 for Trump, two for Kasich, and one each for Cruz, Bush and Rubio.

9:36 p.m. Ben Carson didn't even stay in New Hampshire for primary night and instead got an early start on the South Carolina primary. He's sitting at the bottom of the polls in New Hampshire, with just 2 percent. His campaign circulated this statement in lieu of an appearance:

"One of the most enjoyable aspects of the election process is getting to meet so many great people across the country, and New Hampshire has been no exception. I'm honored and humbled to have the generous backing of so many citizens of The Granite State, and I thank everyone for their kind hospitality and support. I commend the many staff and volunteers from New Hampshire who worked tirelessly to spread our call to return faith, integrity and common sense to American leadership. Across New Hampshire I heard from people who were sick and tired of D.C. insider political games and manipulation. Voters around the country are looking for new leadership in Washington, which is why I'm working so hard to return 'We the People' to the White House. I will carry on this fight for as long as the people stand with me. As we now move on to South Carolina, Nevada and the Super Tuesday states, I will continue to stand for integrity, accountability and honest leadership."

9:28 p.m. It turns out the Republican debate last Saturday mattered.

Almost 70 percent of voters said the debate played an important role in their voting decision and almost 90 percent said it was a factor in their decision (just 8 percent said it was not a factor). Among those who said it was important, Kasich came in second behind Trump with 17 percent of the vote, followed by Bush and Cruz who tied with 12 percent each. Similarly, among those who said the recent debate was a factor in their decision, Kasich came in second.

It seems that voters not only tuned in, but agreed with most pundits who said that Saturday night Kasich and Bush gave their strongest debate performances yet. The late debate and the stock GOP voters put into it may also help explain in part by Rubio was unable to build on the momentum he had coming out of Iowa. According to exit polls, those who thought the debate was important and those who said it was a factor in their decision were less likely to support Rubio than Kasich.

9:25 p.m.Republican voters are nearly evenly split between those who want a candidate with political experience (45 percent) and those who want a political outsider (50 percent). When you dig into the numbers, however, there is a clear divide. Those voters who support candidates from the "establishment" lane support Kasich (28 percent) and Bush (20 percent), while those preferring a nominee from outside the political realm, support Trump (61 percent).

9:17 p.m. CBS News estimates that John Kasich will come in second to Donald Trump in the GOP primary.

"Governor Kasich is now the leading governor in the race and the only one with a realistic chance at the nomination," Kasich senior strategist John Weaver said in a statement. "He showed that a conservative with a positive message will succeed and, in fact, that's the only way for Republicans to win the White House. As the governor of Ohio he knows how to do it."

What a night! Thx to our incredible NH supporters! Time to trade our snow boots for flip flops. See you tmrw SC! https://t.co/dy8KnZ122T — John Kasich (@JohnKasich) February 10, 2016

Kasich's campaign said he will leave for South Carolina Thursday night, and that he is now poised to "accumulate delegates in the early primary calendar before winning Michigan and taking the nomination battle to the Midwest, the home turf where he will win the nomination."

9:13 p.m. Despite finishing far outside of the top three candidates in New Hampshire, Carly Fiorina told supporters Tuesday, "We are going to keep going."

Fiorina added her supporters have "given us wind in our back." With 24 percent of votes counted, Fiorina is in second to last place with just 4 percent of the vote.

8:44 p.m. Some more analysis from the exit polls:

When asked about their view of the way the federal government is working, 48 percent of Republican New Hampshire primary voters were dissatisfied, but not angry, and 40 percent were angry. Of those who said they were dissatisfied or angry, 32 percent voted for Trump followed by 15 percent for Kasich.

Almost half of Republican primary voters (46 percent) said they feel betrayed by politicians from the Republican Party and of those, 32 percent backed Trump. Trump took the largest share of support from both men (34 percent) and women (29 percent). When asked about the most important issues facing the country, economy was seen as the most important. Of those who see that as the most important issue, 27 percent voted for Trump and 24 percent for Kasich.

Trump is also the top candidate choice for the other three issues: Immigration (50 percent), Terrorism (27 percent), and government spending (31 percent).

Trump did very well among Republican New Hampshire primary voters who want to see illegal immigrants deported: 46 percent voted for him. Of the 66 percent of Republican voters who support a temporary ban on Muslims entering the U.S, 42 percent back Trump. About a quarter of those who oppose a ban voted for Kasich.

8:40 p.m. Trump had a strong showing among late deciders as well as first-time primary voters. He also did better than expected among those voters who consider electability as key factor in their voting decision.

LATE DECIDERS FOR TRUMP AND KASICH: John Kasich and Donald Trump both appear to have benefitted from the more than 45 percent of Republican primary voters who chose a candidate at the last minute. Among those who decided in the last few days about 21 percent broke for Kasich while Trump received 20 percent of the vote.

FIRST TIME VOTERS SMALL IN NUMBER, BUT TRUMP WINS THEM: Only about 14 percent are voting in a Republican primary for the first time. Thirty-four percent voted for Trump -- but he also won 31 percent of voters who have previously voted in a primary.

TRUMP HAS A STRONG SHOWING AMONG THOSE WHO CONSIDER ELECTABILITY A KEY FACTOR: Only 12 percent of GOP primary voters focused on electability as a key factor in making their decision today. While it isn't a critical factor among most Republicans, among those who place a high value on it Rubio won with 33 percent of the vote, followed by Trump with 29 percent of the vote.

In Iowa, Rubio also won those voters who most valued a candidate who could win in November. GOP voters are far more interested in having a nominee who shares their values (35 percent), is capable of bringing about change (29 percent) or one who tells it like it is (22 percent). Trump was the top choice of both voters who want a change candidate and those who want a candidate who speaks his or her mind.

8:22 p.m. John Kasich is now leading the battle for second place in the New Hampshire Republican primary.

8:20 p.m. Trump's victory in New Hampshire can be explained in part by the fact that he won almost every key segment of GOP primary voters including independents, conservatives, and moderates. He also nearly tied Cruz when it comes to support among white evangelicals.

INDEPENDENTS: Independents are about 42 percent of the New Hampshire GOP primary electorate today, lower than the 47 percent in 2012, and slightly higher than the 37 percent in 2008. In Iowa last week, just 20 percent were independents. Trump won this group with 32 percent of the vote, followed by Kasich with 18 percent of the vote.

CONSERVATIVES/MODERATES: New Hampshire Republican voters are conservative, but just about 27 percent identify as very conservative -- far fewer than the 40 percent who did so in the Iowa caucus. Trump won the backing of those who describe themselves as very or somewhat conservative (65 percent), followed by Cruz (34 percent) and Rubio (28 percent). Almost three quarters of Republicans (72 percent) identify as conservative this year, up from 53 percent in 2012.

Interestingly, Trump also won moderates voters who make up just more than a quarter of the New Hampshire GOP. Results show that moderates broke for Trump (30 percent) and Kaisch (27 percent).

EVANGELICALS: Evangelicals are less of a factor in New Hampshire than they were last week in Iowa. White evangelicals made up about 23 percent of New Hampshire voters today (compared to 62 percent in Iowa). Whereas Cruz won this group in Iowa, he only did one percent better than Trump among this group of voters in New Hampshire Tuesday (Cruz received 24 percent of their votes and Trump 23 percent).

8:00 p.m. CBS News projects that businessman Donald Trump will win the New Hampshire Republican primary and Sen. Bernie Sanders, I-Vermont, will win the Democratic primary.

There is now a four-way tie for second place between John Kasich, Jeb Bush, Marco Rubio and Ted Cruz.

All polls in the state are now closed.

7:30 p.m. Voters in New Hampshire are casting their ballots before the last polling stations in the state close at 8 p.m. ET. If polls are correct, businessman Donald Trump is poised to sweep the New Hampshire primary with a clear lead over his rival Republicans.

Trump has a double-digit lead over his rivals in most polls, making the most interesting race of the night who will come in second place. There is a tight race between several candidates who would be considered part of the GOP establishment, including Ohio Gov. John Kasich, Florida Sen. Marco Rubio and former Florida Gov. Jeb Bush.

Texas Sen. Ted Cruz - who won the Iowa caucus a week ago - and New Jersey Gov. Chris Christie are also vying to stay in the mix, as well as former Hewlett-Packard CEO Carly Fiorina.

CBS News exit polls show that nearly half of New Hampshire Republican primary voters -- 46 percent -- made their decision on who to vote for within the last few days. About one-fifth, 22 percent, say they made their decision Tuesday.

One exit poll statistic that probably spells good news for a big Trump victory is this: Two third of Republican primary voters, 66 percent, support a temporary ban on Muslims entering the U.S. - something Trump proposed in December.

About half (48 percent) of Republican primary voters want a candidate with some political experience, while the other 47 percent would like the next president to be outside the political establishment. In Iowa, the majority of those who sought an outsider supported Trump. Of those seeking a candidate with experience, Iowa voters supported Rubio and Cruz.

Thirty percent of voters rate the economy and jobs as their top issue, more than those who said it was government spending (27 percent) and terrorism (25 percent).

And one issue which has been a point of contention within the race for many of the GOP candidates, especially for Marco Rubio, ranks much lower as a concern: only 15 percent of GOP respondents chose immigration as the most important issue.

CBS News Poll Analysts Melissa Herrmann and Jeanne Zaino contributed to this story.