Last Updated Feb 24, 2016 1:34 AM EST
1:32 a.m.When he came out on stage to speak to supporters in Nevada, Cruz didn't exactly concede that Trump had won the state -- they were still counting ballots, he noted -- but did congratulate Trump on having a "strong evening."
He continues to argue that his is the only campaign to have beaten Trump and won one of the critical first three primaries.
- Campaign 2016: The delegate scorecard
- Live: Nevada Republican primary results
- Republican nomination calendar
"History teaches us that nobody has ever won the nomination without winning one of the first three primaries. And there are only two people who have won one of the first three primaries: Donald Trump, and us," Cruz said. "The undeniable reality that the first four states have shown is that the only campaign that has beaten Donald Trump and the only campaign that can beat Donald Trump is this campaign."
He almost seemed like he wasn't going to even acknowledge Rubio, who has not placed higher in both South Carolina and Nevada. But then Cruz said, "You can choose between two Washington dealmakers or one proven, consistent conservative."
He also raised the stakes for next week, saying Super Tuesday will be "the most important night of this campaign. Among the states that vote is his home state of Texas, and Cruz closed by noting that tonight he will sleep in his own bed for the first time in a month.
"I cannot wait to get home to the great state of Texas," he said.
1:05 a.m. A triumphant Trump said he and his supporters would be "celebrating for a long time tonight" after his dominant win in the Nevada caucus.
"We're winning winning winning the country. And soon the country's going to start winning winning winning," Trump said.
He ticked off upcoming primary states where he said he has "great numbers": Texas, Tennessee, Georgia, Arkansas, Florida and Ohio. He noted that in Ohio polls, he's even beating Gov. John Kasich, who is also running for president.
"It's going to be an amazing two months," Trump declared, before predicting that "we might not even need the two months" to secure the nomination.
His discussion of the demographics where he performed well evoked an interesting line.
"We won with poorly educated, I love the poorly educated," Trump said. He won a majority of voters, 51 percent, who have no college degree.
12:58 a.m. Despite a distant fourth-place finish, neurosurgeon Ben Carson has no plans to get out of the race.
Carson told supporters following the Nevada caucuses that he believes "things are starting to happen here," and said he thinks the pundits who are already calling the race for the GOP nomination are wrong.
"The interesting thing about my life is I've had so many situations where people have said this is impossible, it can't be done," Carson said. "So unless I hear those words, I don't get excited."
12:38 a.m.Cruz performed best among those who say they are very conservative; getting 31 percent of that vote, but Trump did better and got 38 percent of votes.
Rubio, on the other hand, did best among those who say they were somewhat conservative, winning 27 percent of their votes. Once again, however, Trump did best, and won 50 percent of those votes.
Trump also led among voters who call themselves moderate, getting a majority -- 55 percent -- of their votes.
Forty-one percent of born-again or evangelical Christians picked Trump, and 50 percent of Christians who are not born again or evangelical supported the businessman as well. That's bad news for Cruz, who needed to do well among the most conservative caucus attenders and evangelical Christians. He actually came in third among both groups: Among evangelical and born-again Christians, Rubio got 25 percent and Cruz got 24 percent. Among other Christians, Rubio got 26 percent and Cruz got 18 percent.
12:06 a.m.CBS News projects Donald Trump will win the Nevada GOP caucus based on CBS entrance polling and precinct sampling.
Some good news for Rubio: 24 percent of caucus attenders said the most important quality they are looking for is someone who can win in November -- that's higher than in previous contests this year. And Rubio runs strongest among this group, winning 51 percent support among those voters. Rubio also picks up strength among the late deciders. However, most (76 percent) had decided more than a week ago. Among those who decided in the last few days: Rubio 39 percent; Cruz 23 percent; Trump 22 percent.
Rubio's best strategy might be to convince Republican voters that he can win the general election and Trump cannot.
11:50 p.m. Here's what Republican voters have to say about why they are supporting their respective candidates.
Donald Trump supporters like that their candidate tells it like it is:
Ted Cruz supporters say he shares their values:
And those who support Marco Rubio believe he is electable:
11:27 p.m. Some fresh data from the entrance polls of Nevada Republicans:
Unlike Iowa and South Carolina, most caucus attenders do not consider themselves born-again or evangelical Christians. Sixty-one percent identify as not born-again or evangelical, while just 39 percent do.
Additionally, the conflict over the Supreme Court vacancy caused by the death of Antonin Scalia does not appear to be a deciding factor for most Republican caucus attendees. Just 18 percent said it was the most important factor, while 60 percent said it was one of several factors. Nine percent said it was a minor factor and 5 percent said it was not a factor at all.
10:40 p.m. The Nevada Republican Party also sought to clarify the rules after reports that volunteers at caucus sites are wearing gear to indicate their support for a particular candidate.
It's not against the rules for volunteers to wear candidate gear. Volunteers went through extensive training & are doing a great job— Nevada GOP (@NVGOP) February 24, 2016
10:38 p.m.Most of the early caucus attenders say they had made up their minds at least a week or more ago -- only about a quarter say they made up their minds in the last few days.
10:33 p.m. Nevada Republican voters are the angriest about the federal government compared to Republicans who have caucused or voted in Iowa, New Hampshire and South Carolina:
And the largest percentage of voters surveyed said they are looking for a candidate who shares their values:
10:32 p.m. Caucus attendees are overwhelming older (over 45), they are split between those with and without college degrees, and most do not describe themselves are born-again or evangelical Christians.
10:30 p.m. After some reporters were tweeting about confusion and possible problems at voting sites, the Nevada Republican Party took to Twitter to say everything is fine.
There have been no official reports of voting irregularities or violations. #nvgopcaucus— Nevada GOP (@NVGOP) February 24, 2016
10:25 p.m. Very early entrance poll results offer a glimpse of what is on voters' minds Tuesday night as they make their picks for the GOP nominee.
Asked to best describe their feelings about the way the federal government is working, a majority of Republican caucus attenders say they are angry.
A majority of caucus attendees, 30 percent, say the economy/jobs are the most important issue. But significant numbers of voters also point to government spending (26 percent), immigration (20 percent) and terrorism (20 percent).
More attendees would like the next president to be from outside the political establishment rather than having experience in politics.
8:40 p.m. Republican voters in Nevada are beginning to convene for the presidential caucuses, the fourth GOP presidential nominating contest this year. Caucuses start between 5 and 7 p.m. PT, or 8 and 10 p.m. EST, and they're slated to end at 9 p.m. PT or midnight EST.
Polling suggests GOP front runner Donald Trump could claim his third victory, after winning the New Hampshire and South Carolina primaries; however, the Nevada caucuses have been historically unpredictable.
Unlike the process for the Democrats, Republican voters in Nevada's GOP caucuses will mark their preferred candidate on a secret ballot, and they don't have to stay for speeches. But the Nevada GOP allows participation over a long time period, up to a four-hour window in some counties.
Nevada has 30 delegates available for the Republican National Convention, and they're distributed proportionally based on the results of Tuesday's primary.
After Tuesday, the candidates will face off at the next GOP presidential Republican debate on CNN Thursday and then they'll keep preparing for Super Tuesday next week in which 11 states will hold primaries and caucuses.
CBS News Election Analyst Stanley Feldman contributed to this story.