Republican Donald Trump and Democrat Hillary Clinton took big steps toward securing their parties' presidential nominations on Tuesday with a series of state-by-state victories, but their rivals vowed to keep on fighting.

On Super Tuesday, the 2016 campaign's biggest day of nominating contests, Trump, 69, and Clinton, 68, proved themselves the undisputed front-runners.

Now they are under pressure to show they can unify voters in their respective parties and avoid a potentially disastrous split in their ranks that could hurt them in the November 8 election.

US networks projected Trump won seven states with victories stretching into the deep South and as far north as Massachusetts, adding to a sense of momentum he had built last month by winning three of the first four contests.

Clinton's victories in seven states were just as impressive but in many ways predictable, propelled by African-American voters in southern states like Arkansas, where she and former President Bill Clinton began their political careers.

Trump's rivals Ted Cruz, a US senator from Texas, and Marco Rubio, a US senator from Florida, emphasised their determination to remain in the race.

Cruz, 45, won his home state of Texas and neighboring Oklahoma, bolstering his argument he had the best chance to stop the brash billionaire. Rubio, favourite of the Republican establishment, was projected the winner in Minnesota, his first victory.

Clinton's rival Bernie Sanders, a democratic socialist US senator from Vermont, also won his home state along with Colorado, Minnesota and Oklahoma and vowed to pursue the battle for the nomination in the 35 states yet to vote. He lost to Clinton in Massachusetts, a fifth state he had hoped to win.

Super Tuesday was the biggest single day of state-by-state contests to select party nominees for the November 8 election to succeed Democratic President Barack Obama.

TRUMP WAVES OFF CRITICISM

At a news conference in a chandeliered ballroom at his seaside Mar-a-Lago resort in Palm Beach, Florida, Trump, who has never held public office, dismissed furious criticism aimed at him by establishment Republicans.

Faced with a party in turmoil over his ideas to build a wall between the US and Mexico, deport 11 million illegal immigrants and bar Muslims from entering the country, Trump declared he had expanded the party by drawing in disaffected blue-collar Democrats who like his tough-on-trade rhetoric.

"I am a unifier," he said. "I would love to see the Republican Party and everybody get together and unify and when we unify, there's nobody that's going to beat us."

The rivals of both Trump and Clinton aim to knock them off their pedestals in contests ahead in Michigan, Florida and Illinois.

Trump waved off criticism from the country's top two elected Republicans, House Speaker Paul Ryan and Senate Majority Leader Mitch McConnell, over his delayed disavowal of an endorsement by David Duke, a former leader of the Ku Klux Klan, a white supremacist group.

"I've disavowed," Trump said. "I'm going to get along with Congress, okay? Paul Ryan, I don't know him well, but I'm sure I'm going to get along great with him. And if I don't, he's going to have to pay a big price, okay?" Trump said in remarks that could further inflame party tensions.

Clinton, who still faces a well-funded Sanders despite having taken control of the Democratic race, was eager to assail Trump as a way of getting Democratic voters used to the idea of her as the nominee.

"The stakes in this election have never been higher and the rhetoric we're hearing on the other side has never been lower," Clinton told supporters in Miami. "Trying to divide America between us and them is wrong, and we're not going to let it work."

'DONALD TRUMPS OF THE WORLD'

Sanders thanked cheering supporters in his hometown of Burlington, Vermont, and assailed the Republican front-runner.

"We are not going to let the Donald Trumps of the world divide us," said Sanders, 74, adding that he expected to pile up "hundreds" of convention delegates in voting on Tuesday.

For Rubio, 44, it was a day of reckoning. His losses piled up after a week in which he labeled Trump a "con artist" and exchanged schoolyard taunts with him over, for example, who wet their pants under pressure at a debate last week.

Suddenly, the contest on March 15 in Florida, his home state, loomed over him as a must-win state.

"Florida, I know you're ready," Rubio said. "The pundits say we're underdogs, I'll accept that. We've all been underdogs."

Rubio's plight was such that Senator Lindsey Graham, an establishment South Carolina Republican, told CBS News that Republican voters might need to rally around Cruz, who has been one of the most disliked public figures in Washington.

"I can't believe I would say yes, but yes," Graham said when asked about the idea of supporting Cruz as a way of stopping Trump.

Cruz told supporters at his victory party in Texas that Trump was a "Washington dealmaker, profane and vulgar, who has a lifelong pattern of using government power for personal gain."

The networks had yet to project a winner for Republicans in Alaska. - Reuters