AT least 120 million Americans took to the polling booths in record numbers as the longest, costliest and most intense presidential campaign finally came to a close with all eyes on Ohio and eight other swing states.

Lengthy waits at polling stations were reported in the state, which along with Florida and Virginia were key barometers of the nation's mood as it decided whether President Barack Obama should be given a second term or Republican Mitt Romney should be their commander in chief.

People turned out in force in New Jersey and New York, which bore the brunt of superstorm Sandy last week, undeterred by the cold weather and the makeshift nature of some polling stations.

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Some voters were forced to wait more than three hours in the densely populated Miami area, the same state where the local Democratic Party launched legal action over queues of up to seven hours at the weekend – claiming it will deter people from casting their ballot.

In New Orleans, people queued up at a fire station to choose their next president and a seven-seater bicycle was used to ferry students out at George Mason University campus in Fairfax, Virginia.

Barack Obama stopped off in a campaign office in his home city of Chicago to help in a last-minute telephone campaign to get Democrat voters to the polls. He congratulated Mr Romney on "a spirited campaign", saying: "I know his supporters are just as engaged and just as enthusiastic and working just as hard today."

After a hard-fought and sometimes bitter contest, which began nearly two years ago and has cost at least $2 billion, Mr Obama appeared optimistic; two national polls gave him a three-point lead over his rival and he was also narrowly ahead in most swing states.

He said: "We feel confident we've got the votes to win but it's going to depend ultimately on whether those votes turn out. So I would encourage everybody on all sides just to make sure you exercise this precious right that we have, that people fought so hard for us to have."

Thanking volunteers for their efforts, he added: "I expect that we'll have a good night but no matter what happens, I just want to say how much I appreciate everybody who supported me, everybody who's worked so hard on my behalf." The sports-mad president was due to play some basketball to wind down before the long nervous wait.

Mr Romney was asked by journalists how it felt to see his name on the ballot paper after he attended his local polling station in Boston. He said: "We've been working for this a long time and to be on the ballot for the President of the United States is very humbling."

He made one more trip to Ohio, which some Democrats branded an act of desperation.

However, minutes after Mr Romney touched down on the tarmac in Cleveland, Joe Biden, the Vice-President, swooped in on Air Force Two for an unannounced – but what the White House insisted was a long-planned – stop-off. The move by Mr Obama's running mate appeared to be a concerted bid to dampen Mr Romney's last-gasp visit to the bellwether state.

The ex-governor of Massachusetts, who was waiting for his running mate Paul Ryan to land, stayed onboard his plane until Mr Biden boarded his motorcade and left the tarmac, as Mr Ryan's plane landed minutes later.

The awkward three-way stop underlined the importance of the must-win Ohio.

"That sums up what is going on today. Everyone is competing for that last vote," said Stephanie Cutter, deputy manager for the Obama campaign.

The midwestern state, which has 18 electoral college votes, is seen as the most crucial in the candidates' race for the 270 votes needed to win the presidency.

US TV networks agreed to withold the announcement of their exit polls until all the voting in each state had been completed. This followed a debacle in 2004, when they favoured Democrat John Kerry, but in fact it was George W Bush who came through to win a second term. There were also fears that people would be discouraged from voting if they believed a state had been won The channels also kept exit polls a closely guarded secret in case they were tweeted or posted on Facebook.

The closeness of the race meant each side had teams of lawyers on hand to make sure nothing untoward happened at the polling stations. Problems were reported at several across the country. A confrontation erupted in Pennsylvania involving Republican inspectors over access to some of the booths while there was a late court fight in Ohio over election software.

Voters were left confused over the ID they needed to cast their ballots in Pennsylvania. Barbara Arnwine, of the Lawyers' Committee for Civil Rights said: "Poll workers have been poorly and wrongfully trained, and they are standing there and sitting there and requiring people to show ID, and sending people home if they don't have the ID.

A judge last month blocked the state from requiring voters to show photo identification, a setback for Republican state officials who had championed the law.

Zack Stalberg, president of the Committee of Seventy, a Philadelphia-based elections watchdog, said it had taken hundreds of calls from confused voters. He added that up to 10% had been turned away or seen others refused entry to polling booths because of a lack of photo ID.

Meanwhile, poll organisers faced criticism for the lengthy queues. "When you look at the lines that have formed in places like Ohio, they are longer than the lines in Baghdad and Kabul," said Wade Henderson, president of the Leadership Conference on Civil and Human Rights.