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The contest in Ohio is seen as too close to call
Democratic front-runner Barack Obama says rival Hillary Clinton may have to quit if he wins polls in Texas and Ohio. She says she will carry on.
Contests in two smaller states also voting - Vermont and Rhode Island - may prove crucial in such a tight race.
On the Republican side, John McCain - already comfortably ahead of his rivals - could seal the nomination.
But most attention is focused on the Democrats' battle.
For the Democrats, Texas is the biggest prize remaining, with 228 delegates to the nominating party convention in August up for grabs.
Voting in the Democratic polls is proportional, meaning that Mrs Clinton needs landslide victories on Tuesday and beyond to catch up with Mr Obama.
An opinion poll published as voting began gave Mrs Clinton a narrow lead in Texas - though within the poll's margin of error. It had the two candidates exactly level in Ohio.
"I feel really good about today," Mrs Clinton told voters in Houston. "I think it's going to turn out well."
She added: "You don't get to the White House as a Democrat without winning Ohio."
Since the nationwide Super Tuesday contests on 5 February, Mr Obama has won 11 states in succession and leads Mrs Clinton in the delegate count.
Mr Obama seemed confident that he would win in the end.
"We've got a very sizeable delegate lead that is going to be hard to overcome," he told reporters travelling with him.
"You'll recall that when we were running those 11 races in a row the theory was they had to blow us out in Texas and Ohio.
"I don't think that's going to happen."
Mr Obama has spent twice as much as Mrs Clinton on TV adverts in the state, including a number in Spanish. Hispanics account for about one in five eligible voters in Texas.
- Hillary Clinton
13 states, 1,276 delegates
- Arizona, Arkansas, California, Florida, Massachusetts, Michigan, Nevada, New Hampshire, New Jersey, New Mexico, New York, Oklahoma, Tennessee
- Barack Obama
24 states, 1,386 delegates
- Alabama, Alaska, Colorado, Connecticut, Delaware, Georgia, Hawaii, Idaho, Illinois, Iowa, Kansas, Louisiana, Maine, Maryland, Minnesota, Missouri, Nebraska, North Dakota, South Carolina, Utah, Vermont, Virginia, Washington state, Wisconsin
2,025 delegates needed for nomination. Source AP (includes all kinds of delegates)
Q&A: US election delegates
- Mike Huckabee
8 states, 257 delegates
- Alabama, Arkansas, Georgia, Iowa, Tennessee, West Virginia, Kansas, Louisiana
- John McCain
17 states, 1014 delegates
- Arizona, California, Connecticut, Delaware, Florida, Illinois, Maryland, Missouri, New Hampshire, New Jersey, New York, Oklahoma, South Carolina, Vermont, Virginia, Washington state, Wisconsin
- Mitt Romney
11 states, 255 delegates
- Campaign suspended
Alaska, Colorado, Maine, Massachusetts, Michigan, Minnesota, Montana, Nevada, North Dakota, Wyoming, Utah
1,191 delegates needed for nomination. Source: AP (includes all kinds of delegates)
According to exit polls for the Associated Press news agency, however, they have cast nearly a third of the election day votes in the state - up from a quarter in 2004. In previous contests this year, they have favoured Mrs Clinton.
African-American voters - who have heavily supported Mr Obama - accounted for about a fifth of the votes cast in Texas, exit polls indicate, about the same as four years ago.
The economy was the most important issue for Democratic voters in all four states, especially Ohio, exit polls suggest. The Iraq war came a close second in Vermont.
For the Democrats, a total of 370 delegates are at stake in the four races, which includes 67 delegates up for grabs in Texas caucuses, which begin after the day-long primary vote.
Mr Obama currently has 1,386 delegates to Mrs Clinton's 1,276, according to the AP. A total of 2,025 is needed to secure the Democratic Party's nomination.
The New York senator and former first lady has played down suggestions that she is facing a make-or-break moment.
Rhode Island 0200GMT
Texas 0200GMT - caucuses then begin
The BBC's Kevin Connolly, in Ohio's state capital, Columbus, says the struggle between the two senators remains fierce and close, and it is far from certain that America will get the clear outcome from these latest battles that it craves.
Mrs Clinton has been focusing her attacks on Mr Obama's foreign policy and national security experience, echoing a campaign advert asking who would respond better to a national emergency in the middle of the night.
Mrs Clinton insists she can go all the way to the White House
Mr Obama, senator for Illinois, has countered that with an advert questioning Mrs Clinton's judgement in supporting the invasion of Iraq from the start.
Meanwhile, Mr Obama denied on Monday that his campaign had privately assured Canada his criticism of the North American Free Trade Agreement (Nafta), widely opposed in economically depressed Ohio, was just for political show.
A leaked memo suggested Mr Obama's senior economic adviser, Austan Goolsbee, had given Canadian officials the impression that Mr Obama's criticism over the US free trade deal with Canada was "political positioning".
John McCain is expected to beat closest rival Mike Huckabee
The dispute led Canadian Prime Minister Stephen Harper to assure parliament that the memo leak was not an attempt to scupper Mr Obama's chances or to favour Mr McCain, who strongly backs Nafta.
On the Republican side, Mr McCain is expected to beat his chief rival Mike Huckabee in all four states.
The Arizona senator currently has 1,014 delegates, according to the AP, while Mr Huckabee has 257.A total of 1,191 delegates is needed to claim the nomination at the party's national convention in September. If you enjoyed this post Subscribe to our feed