The focus of the US presidential race is turning to the southern states, with key primaries in South Carolina and Florida in the next 10 days.
The Democrat and Republican races remain wide open, with top candidates hoping to gain momentum.
Democrat Hillary Clinton, after winning Nevada, faces a hard struggle in South Carolina, where Barack Obama may be helped by the large black population.
Republican John McCain faces a strong Florida challenge from Rudy Giuliani.
The former New York mayor is concentrating on larger states, but currently trails Mr McCain in polls in Florida, which votes on 29 January.
The BBC 's Jonathan Beale in Washington says that if Mr McCain can win in the conservative South, he has a chance to build real momentum.
The Florida and South Carolina ballots precede Super Tuesday on 5 February, when 22 states will hold nomination contests.
On Saturday, Mr McCain saw off a close challenge by Mike Huckabee in South Carolina to win by 33% to 30%.
"I think we're obviously doing very well," Mr McCain told reporters after the victory. "This is still very competitive."
South Carolina is seen as significant because, since 1980, every Republican candidate who has won in South Carolina has gone on to win the party's nomination.
Mr Giuliani said he was ready for the battle.
"The case for me is that I am the strongest fiscal conservative," he said, adding that unlike some other candidates he had "executive experience".
In Saturday's other Republican contest, Mitt Romney won Nevada's caucus, taking 51% of the vote.
Mrs Clinton beat Barack Obama by 51% to 45% in Nevada, based on 98% of returns.
Mrs Clinton described her win as "an especially wonderful day for me."
The Associated Press reported that Mr Obama had won 13 delegates to Mrs Clinton's 12, partly because Mr Obama performed strongly in rural areas where a small lead is more likely to yield an extra delegate.
Going into the Nevada vote, Mrs Clinton was backed by influential politicians in the state's Hispanic community, which makes up about 25% of the population, while Mr Obama had the support of a powerful local union organisation.
Mrs Clinton and Mr Obama will now take their struggle for the Democratic nomination to South Carolina, where the party's primary takes place on 26 January.
Correspondents say the key issue will be which of the two candidates can attract the support of the state's black community, who make up a third of the population.