Republican presidential candidates have taken part in a fiery debate as they clashed over foreign policy and immigration.

The rivals traded blows, with insults shouted across the stage amid booing from the audience and a warning the debate was being "dragged into the dirt".

Billionaire Donald Trump leads the polls ahead of next week's South Carolina primary vote, having won New Hampshire last Tuesday.

Trump repeatedly called his opponents liars and interrupted them, saving his most acrimonious attacks for former Florida Governor Jeb Bush.

Trump blasted Bush's brother, George W. Bush, for launching the Iraq war in 2003, describing it was a "big, fat mistake". He also noted that the 9/11 attacks had occurred on Bush's watch.

When members of audience booed Trump, he dismissed them as "lobbyists and special interests" supporting Bush.

"I'm sick and tired of him going after my family," Bush responded. "My dad is the greatest man alive in my mind. While Donald Trump was building a reality TV show, my brother was building a security apparatus to keep us safe. And I'm proud of what he did."

Bush added: "He had the gall to go after my mother. My mother is the strongest woman I know."

Trump hit back: "She should be running."

Ted Cruz was called a liar by Trump when he said he would appoint liberals to the Supreme Court

Cruz and Marco Rubio renewed their battle over who is toughest on illegal immigration, with Cruz insisting the Florida senator was for "amnesty" but now is against it for political purposes.

He insisted that Rubio had said in Spanish on Univision that he would not rescind an executive order signed by President Barack Obama in support of the children of illegal immigrants.

Rubio shot back: "I don't know how he knows what I said on Univision because he doesn't speaking Spanish."

Cruz then began speaking in Spanish across the stage.

Ohio Governor John Kasich said the Republicans were "fixing to lose the election to Hillary Clinton" if they didn't stop bickering with each other.

Retired neurosurgeon Ben Carson was overshadowed in the debate, at one point joking about his lack of involvement.