While Mr Giuliani leads in national polls at present, he trails Mr Romney in Iowa and in New Hampshire, which host respectively the first caucus and the first primary.
Sanctuary claims
Illegal immigration is a hot topic for Republican voters, many of whom want tighter controls, and the candidates were forced to confront it straight away.
Mr Romney provoked Mr Giuliani's wrath when he suggested Mr Giuliani had created a "sanctuary city" for illegal immigrants as mayor of New York.
Mr Giuliani then accused Mr Romney of employing illegal immigrants at his mansion - creating a "sanctuary mansion".
Mr Romney - who hired landscapers at his Massachusetts home who turned out to be in the country illegally - retorted that it would "not be American" to check the papers of contractors simply because they had a "funny accent".
The candidates faced public questions on topics including abortion, gun control, gays in the military, and God.
Sen John McCain, chief proponent of the recent troop "surge" in Iraq, took on Ron Paul, whose libertarian outlook and opposition to the Iraq war has won him a significant following on the internet and campaign pledges of millions of dollars.
Sen McCain accused Mr Paul of promoting the "kind of isolationism... [that] caused World War II".
Mr Paul hit back, asking: "The real question you have to ask is why do I get the most [campaign] money from active-duty officers and military personnel?"
Race to the right
For several candidates an evident concern was to burnish their right-wing credentials.
Fred Thompson used video clips of two rivals to question their right-wing leanings, while Mr Romney conceded he had been "wrong" when he previously held a pro-choice stance on abortion.
The BBC's Jane Little, who watched the debate, says the main laugh of the night was won by Mike Huckabee, former Baptist minister and governor of Arkansas.
He side-stepped a question on the death penalty - "What would Jesus do?" - by replying: "Jesus was too smart to run for public office".
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