Donald Trump continued to dominate the Republican race for US president by causing further controversies in the first debate.
The tycoon's refusal to rule out a third-party run drew boos from the audience on Thursday night.
And he stood by offensive comments he has made in the past about women.
The 10 candidates in Cleveland, selected by Fox News on the basis of recent national polls, provided a frank and bruising exchange of views.
Facing some tough questions from the debate moderators, the candidates tried to stand out in a crowded Republican field.
Mr Trump stumbled on his past support for a national healthcare system but his most uncomfortable moment came when moderator Megyn Kelly challenged him on his views about women.
"You've called women you don't like fat pigs, dogs, slobs and disgusting animals," she said.
He answered by joking that he only said that about actress Rosie O'Donnell and stating that political correctness was one of the country's biggest problems.
The Fox News moderators were widely praised for their tough questions
The crowd became hostile when Mr Trump said he would run as an independent, an admission that enraged Mr Paul. "He buys and sells politicians of all stripes," said the senator.
One of the loudest rounds of applause was for Mr Rubio when he mocked Hillary Clinton, who leads the Democratic field.
"First let me say, I think God has blessed us. He's blessed the Republican Party with some very good candidates. The Democrats can't even find one."
On Thursday, the Democratic Party announced it would hold its first debate in Nevada in October, hosted by CNN.
By next summer, each party will have a presidential nominee who will do battle in the race for the White House. Votes will finally be cast in November 2016.