Last Updated Feb 20, 2016 9:49 PM EST
9:48 p.m. Cruz and Rubio are still locked in a tight race for second place.
9:45 p.m.Ted Cruz spoke with supporters from his South Carolina headquarters Saturday. "Friends, once again, we have made history," Cruz said, noting that "right now we are effectively tied for second place" with rival Marco Rubio.
He addressed Bush's withdrawal from the race at the beginning of his speech, saying the former Florida governor "brought honor and dignity" to the race. Cruz said Bush was "a man who didn't go to the gutter and engage in insults and attacks."
"We don't know the exact results right now," he said, "Right now, we are effectively tied for second place."
9:32 p.m.Speaking to supporters in Wakefield, Massachusetts, Ohio Gov. John Kasich addressed the results coming out of South Carolina's Republican primary Saturday night.
"We came roaring out of New Hampshire," Kasich said, referring to his strong second-place finish behind Donald Trump in the first-in-the-nation primary state. "And tonight one more time, we have defied and overcome expectations in the state of South Carolina."
"For all the people watching -- you fasten your seatbelts, please," he said.
Kasich is trailing in fifth place in South Carolina, with just eight percent of the vote, according to CBS News exit polls. Former neurosurgeon Ben Carson is the only candidate behind him, with seven percent of support.
9:23 p.m. Marco Rubio came out to speak to his supporters even though the race between him and Cruz was still too close to call. They are both battling for second place.
"This has been a three-person race and we will win the nomination," said Rubio, who congratulated Bush on a hard-fought campaign and called him "the greatest governor in the history of Florida."
Rubio said the 2016 campaign has been a "long road" and there were many people on the campaign when it first kicked off.
"Practically speaking, it's down to three," he said, adding that his campaign provides the best chance to unify the GOP and unify the country.
8:54 p.m. Donald Trump addresses his supporters in South Carolina.
"We are going to start winning for our country," Trump said.
"It's tough, it's nasty, it's mean, it's vicious. It's beautiful. When you win, it's beautiful," he added of campaigning.
The businessman said he was headed to Nevada to campaign Sunday and predicted he would do "terrific" both there and in the SEC primary, the collection of southern states that vote on Super Tuesday.
While he congratulated rivals Ted Cruz and Marco Rubio by name, he neglected to mention another opponent: Jeb Bush, who withdrew from the White House race Saturday night.
Trump was joined by several family members on the stage, including wife Melania Trump.
"He will be the best president," Mrs. Trump told South Carolina supporters.
After Trump congratulated rivals Ted Cruz and Marco Rubio for doing "a very good job," the audience booed. But then he promised, "We go back to war tomorrow morning."
The billionaire's victory in South Carolina's GOP primary can be explained in part by the way he edged out his opponents in key demographic groups.
Trump won among men (35 percent) compared to Cruz's 24 percent and Rubio's 22 percent. Interestingly, he also did better than all his opponents among women, winning 28 percent of the vote to Cruz and Rubio's 24 percent respectively.
Three-quarters of those who turned out in the primary were 45 years of age or older. Among these older voters, Trump won 33 percent to Rubio and Cruz who split evenly at 22 percent each.
Another key constituency that helped catapult Trump to victory is those who have less than a college education. Forty-six percent of those who turned out do not have a college degree and of those voters, 40% supported Trump compared with 27 percent for Cruz and 17 percent for Rubio.
Finally, Trump did very well with white evangelicals who make up a key voting group in South Carolina's Republican party. While Cruz was expected to do very well with this group, Trump edged him out 32 percent to 27 percent.
8:40 p.m. Jeb Bush drops out of the White House race.
"I am proud of the campaign that we have run to unify our country and to advocate conservative solutions that would give more Americans the opportunity to rise up and reach their god given potential," he told his supporters after South Carolina's GOP primary. "But the people of Iowa and New Hampshire and South Carolina have spoken and I really respect their decision so tonight I am suspending my campaign."
Bush was joined by South Carolina Sen. Lindsey Graham and Columba Bush on stage in Columbia, South Carolina.
8:28 p.m. More than 70 percent of voters said South Carolina Gov. Nikki Haley's endorsement of Rubio was not an important factor in their vote while a quarter said it was an important factor.
8:21 p.m. GOP voters see Trump as the candidate who is in the best position to handle the economy. Forty-four percent said Trump would handle it best while 20 percent said the same for Cruz and 14 percent for Rubio.
8:00 p.m. Cruz and Rubio are locked in a tight battle for second.
7:50 p.m. About three-quarters of GOP voters described themselves today as born-again or evangelical Christians. Of that group, Trump won slightly more of the vote, 30 percent, than Cruz, who got 29 percent.
Trump also did better than any of the other candidates with those voters who aren't evangelical or born-again.
7:36 p.m. Cruz and Rubio are now battling for second place.
7:30 p.m. CBS News now projects that Trump will win South Carolina's GOP primary.
7:29 p.m. CBS News reports Trump has a narrow edge over Cruz and Rubio.
7:19 p.m. More than half of Republican voters, 53 percent, said they feel betrayed by their own party. Forty-four percent said they don't feel betrayed.
7 p.m.Donald Trump, Ted Cruz and and Marco Rubio are locked in a three-way race, far ahead of the rest of the pack, according to CBS News exit polling.
6:09 p.m. Forty-one percent of voters said Trump is running the most unfair campaign while a third said the same of Ted Cruz.
5:58 p.m. More than half of Republicans said they made up their mind about who they were going to support last month, or even before that. On the other hand, 45 percent said they decided in the last week, 24 percent decided in the last few days and 14 percent made their decision today.
5:52 p.m. More than 95 percent of GOP primary voters said they are either somewhat or very concerned about the future of the nation's economy. Two percent said they are not concerned with the economy.
5:44 p.m. More than half of GOP voters, 53 percent, said they support offering illegal immigrants working in the U.S. the chance to apply for legal status. Forty-three percent, meanwhile, said those who are in the U.S. illegally should be deported and sent back to their country of origin.
5:41 p.m. GOP voters in South Carolina are almost evenly split on whether they prefer a candidate from the establishment vs. a political outsider. Forty-eight percent said they prefer that the candidate has experience and 46 percent said they prefer someone from outside the establishment. This closely mirrors what voter in New Hampshire told pollsters.
5:33 p.m. Three-quarters of GOP primary voters support Donald Trump's proposal to block Muslims from entering the United States. Twenty-three percent, on the other hand, said they oppose the proposed ban.
5:27 p.m. More than half of voters said they are dissatisfied with the way the government is working and 40 percent said they are angry. Only 7 percent said they are satisfied with how the federal government is working.
5:23 p.m. More than three quarters of voters in the South Carolina primary say it matters either a great deal or somewhat that a candidate share his religious values. Less than a quarter, by contrast, say it doesn't matter at all.
5:00 p.m. Early exit polling results is beginning to come in from South Carolina's GOP primary soon. The state's Republican voters have been showing up to their polling sites throughout the day to cast ballots in the GOP's third presidential nominating contest.
Early exit polls in South Carolina suggest that voters see terrorism as the most important issue facing the country (32 percent), followed by those who see the economy (28 percent) and government spending (27 percent) as the most critical issues. In contrast only one out of 10 respondents said that immigration is the most important issue facing the nation.
Voters are choosing among Donald Trump, Ted Cruz, Marco Rubio, Jeb Bush, Ben Carson and John Kasich.
Polls are slated to close at 7 p.m. EST. Meanwhile, the results are coming in quickly from Nevada's Democratic caucuses.
Trump has consistently led the GOP pack in South Carolina, though some recent polls have indicated a tightening race. A Wall Street Journal/NBC News/Marist poll released Friday, for example, found he now only leads his closest rival, Cruz, by five percentage points with 28 percent support. A Bloomberg Politics pollreleased Wednesday found Trump with a 19-percentage-point advantage over Cruz. Thirty-six percent said they back Trump and 17 percent support Cruz.
A CBS News national poll released Thursday found Trump with a 17-percentage-point lead over Cruz.
CBS News Poll Analyst Jeanne Zaino and Reena Flores contributed to this story.