The US presidential rivals have spent a hectic final day of campaigning criss-crossing the country in a last push for votes in key states.

Republican John McCain, trailing in opinion polls, started early in Florida and is to finish in Nevada, more than 1,800 miles (3,000km) west.

Democrat Barack Obama, at his last campaign rally in Virginia, told voters he had one word for them: "Tomorrow."

On Tuesday, voters give their verdict and elect the 44th US president.

On the eve of the ballot, Republican vice-presidential candidate Sarah Palin was cleared by an investigator of abusing her power as governor of Alaska.

The report for the Alaska Personnel Board found that there was "no probable cause to believe that the governor, or any other state official" violated state ethics laws.

An earlier investigation for Alaska's state legislature had found Mrs Palin did abuse her office by allowing her husband and staff members to pressure Alaska's top law enforcement official to fire her former brother-in-law.

Mrs Palin sacked Walt Monegan, the state public safety commissioner, but denied it was because of his refusal to dismiss her sister's ex-husband, a state trooper.

Tributes to grandmother

Meanwhile, Senator Obama said his grandmother, Madelyn Dunham - who largely raised him as a child - had died aged 86 in Hawaii after losing her battle with cancer.

In a joint statement with his half-sister, he described her as "the cornerstone of our family, and a woman of extraordinary accomplishment, strength, and humility", adding that their debt to her was "beyond measure".

Senator McCain and his wife Cindy issued a statement offering their deepest condolences to Mr Obama and his family.

At an event in North Carolina, Mr Obama appeared emotional as 

he spoke of his grandmother, saying she had died peacefully in her sleep with his sister by her side.

Later, giving his final speech of a 21-month-long campaign, he told supporters in Manassas, Virginia, that he had found the long journey to election day both humbling and enriching.

"You have moved me again and again, you have inspired me, sometimes when I am down you've lifted me up," he said.

"You've filled me with new hope for our future and you've reminded me about what makes America so special."

He went on: "We are less than one day away from bringing about change in America."

A USA Today/Gallup poll published on Monday finds likely voters favouring Mr Obama by 11 points over Mr McCain, 53-42%.

Other national polls indicate Mr Obama's lead over his rival is holding steady at between five to 11 percentage points.

But the BBC's James Coomarasamy, in Washington, says that while Mr Obama has held a pretty steady lead for several weeks, there are a number of factors that could undermine the predictions of the pollsters.

Among them, he says, are the role the Illinois senator's skin colour may play in voters' intentions; whether newly-registered voters will actually vote; and the Palin effect - whether Mr McCain's running mate Sarah Palin has energised or alienated Republicans.

McCain defiant

Both camps are keenly aware of the need to get voters out in the states that polls suggest remain in the balance.

Mr McCain was dashing through half a dozen states on the marathon campaign's final day - including Florida, Tennessee, Pennsylvania, Indiana, New Mexico and Nevada - before heading home to Arizona.

Various polls suggest Mr Obama has a two- to four-point lead over him in electoral vote-rich Florida.

On Monday morning, the 72-year-old told a crowd of about 1,100 supporters in Tampa, Florida: "Senator Obama is running to spread the wealth, I'm running to create more wealth."

Mr Obama, 47, spent Monday targeting states that four years ago voted Republican but where he now has a chance of winning, including Virginia and North Carolina, which have not backed a Democratic hopeful in decades.

In Jacksonville, Florida, he reminded his supporters that Mr McCain had told the town on 15 September that "the fundamentals of our economy are strong", only hours before Lehman Brothers went bust and Merrill Lynch was bought by Bank of America.

The crowd jeered at the Republican candidate's statement, but Mr Obama told them: "You don't need to boo, you just need to vote!"

Both campaigns have thousands of volunteers working flat-out manning phone banks, handing out brochures and knocking on doors ahead of Tuesday's election.

Some 130 million Americans are expected to cast a vote, in a higher turnout than in any election since 1960, the BBC's North America editor Justin Webb says.

A record 27 million people had already cast absentee or early ballots as of Saturday night.

Under America's Electoral College system, states are apportioned votes based on their population, the biggest being California with 55 votes.

A candidate needs to gain 270 out of the 538 Electoral College votes to win the presidency.

When Americans go to the polls, as well as choosing a new president and members of Congress, they will be casting votes on a wide range of ballot initiatives such as same-sex marriage, abortion and animal rights.

Polls suggest the six closest state races on election day will be in Florida, Indiana, Missouri, North Carolina, Nevada and Ohio.