Donald Trump has won the South Carolina primary in the Republican race for president, cementing his status as the man to beat for his party’s nomination.
In the Democratic contest, Hillary Clinton beat Vermont Senator Bernie Sanders in a tight race in Nevada.
Both results will be key ahead of the “Super Tuesday” round on 1 March, when a dozen more states make their choice.
The Trump victory claimed a major scalp when former Florida Governor Jeb Bush dropped out of the race.
He finished a distant fourth, days after his brother, former President George W Bush, made a rare political appearance to boost his flagging campaign.
Republican senators Ted Cruz and Marco Rubio were locked in a battle for second place in the state.
Anthony Zurcher, BBC News, Nevada
Hillary Clinton is breathing a sigh of relief, while the Republican faithful wonder where their party is headed.
The former secretary of state has now put a disheartening defeat in New Hampshire behind her. A loss in Nevada could have sent her campaign reeling. Instead she heads to what should be a friendly reception in South Carolina next week with the possibility of starting a winning streak that she could ride to the nomination.
For Republicans, Donald Trump is the one boasting a winning streak. He has a second-place finish in Iowa and double-digit victories in New Hampshire and now in South Carolina.
If this was the CV of any other candidate, the nomination would likely be secure. The prospect of Mr Trump as the Republican standard-bearer, however, may be too much for some in the party to stomach.
Ted Cruz may yet beat the New York billionaire in the South, and Marco Rubio may consolidate establishment support and pull ahead in the winner-take-all states that vote later in March.
With Jeb Bush dropping out, it’s a scenario that could unfold – it is a glimmer of hope, but that’s all mainstream Republicans have at this point.
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“There’s nothing easy about running for president,” Trump said at his victory rally in Spartanburg on Saturday evening.
“It’s tough, it’s nasty, it’s mean, it’s vicious. It’s beautiful – when you win it’s beautiful.”
His supporters erupted at his campaign headquarters when the projected results were announced.
The New York billionaire’s campaign has been dogged in controversy, with his latest spat with the Pope on Christian values hitting the headlines this week.
But it could not stop him winning his second state, to add to his New Hampshire triumph. Texas conservative firebrand Mr Cruz has the only other win, after victory in Iowa.
But many eyes will now be on Florida Senator Marco Rubio, who the Republican mainstream core will likely look to as their leading candidate now that Jeb Bush has quit.
“This country is ready for a new generation of conservatives to lead this country into the 21st century,” Mr Rubio said.
Image copyright Reuters Image caption Mrs Clinton overcame an unexpected strong surge by Senator Bernie Sanders in Nevada
Thousands of miles west, Mrs Clinton narrowly beat Mr Sanders, who had beaten her convincingly in New Hampshire in their last contest.
“Some may have doubted us, but we never doubted each other,” Mrs Clinton told supporters at a victory rally in Las Vegas. “This is your campaign.”
The battle between them has grown increasingly close in recent weeks, with the former secretary of state expected to win Nevada in double digits just weeks ago.
But the Vermont senator, who has successfully galvanised young voters with his calls for free university education, appears to have performed better than expected among Hispanics in Nevada.
According to NBC exit polls, Mr Sanders won among Hispanics with 53% of the vote but lost among black voters earning just 22% of their vote.
Mrs Clinton’s next test will be in the Democrats’ South Carolina primary on 27 February. Republicans will hold their own caucuses in Nevada on Tuesday.
Once each state has voted in the primaries or caucuses, the delegates won by each party candidate will go on to endorse them at party conventions in July.