National polls by Washington Post/ABC News and the Pew Research Centre both give Mr Obama a three-point edge over his rival.
The importance of the crucial swing state of Ohio for both campaigns was underlined by Vice-President Joe Biden, whose plane also arrived in Cleveland on an unannounced stop.
After a hard-fought race that began nearly two years ago and cost more than $2bn (£1.3bn), national polls by Washington Post/ABC News and the Pew Research Center both give Mr Obama a three-point edge over his rival.
As many as 30 million voters have already cast their ballots, with more than 30 states allowing either absentee voting or in-person early voting.
On the stroke of midnight, the first votes were cast and quickly counted in the tiny village of Dixville Notch in New Hampshire. They resulted in a tie with five votes each for Mr Obama and Mr Romney.
Polling stations will begin closing in eastern states at 19:00 EST (00:00 GMT).
'Work not done'
Mr Obama has already voted in his adopted hometown of Chicago, becoming the first sitting presidential candidate ever to vote early. Mr Romney is expected to cast his own ballot in Belmont, Massachusetts, later on Tuesday.
The election is decided by the electoral college. Each state is given a number of electoral votes in rough proportion to its population. The candidate who wins 270 electoral votes - by prevailing in the mostly winner-take-all state contests - becomes president.
Also on Tuesday's ballot are a handful of state governors, one third of the seats in the 100-member US Senate and all 435 seats in the House of Representatives.
Republicans are expected to keep control of the House, while Democrats were tipped to do the same in the Senate.
The presidential candidates spent Monday frantically criss-crossing the crucial battleground states including Ohio, Florida, Iowa and Virginia, making final appeals to voters. Their task: Push their own supporters to the polls while persuading the sliver of undecided voters to back them.
In speeches, Mr Romney kept up his attack on Mr Obama's record, reciting a litany of statistics he says illustrate the president has failed to lift the US economy out of the worst downturn since the Great Depression that followed the stock market crash of 1929.
"If you believe we can do better, if you believe America should be on a better course, if you're tired of being tired... then I ask you to vote for real change," Mr Romney told a rally in a Virginia suburb of the capital, Washington DC.
More than 30 million voters had already cast their ballots before Tuesday's polls opened, with more than 30 states allowing either absentee voting or early voting in person. That is nearly a quarter of the total votes cast in the 2008 presidential election, when more than 130 million people voted.
"We've come too far to turn back now," the president said in Ohio. "We've come too far to let our hearts grow faint... We'll finish what we started. We'll renew those ties that bind us together and reaffirm the spirit that makes the United States of America the greatest nation on Earth."
On the stroke of midnight, the first election day votes were cast and quickly counted in the tiny village of Dixville Notch in New Hampshire. They resulted in a tie with five votes each for Mr Obama and Mr Romney.
In areas of New Jersey and New York that were damaged by storm Sandy a week ago, turnout was described as heavy. One high school being used as a shelter for displaced families doubled up as a voting centre.
Across the US, voting took place against a variety of backdrops - from a launderette and an internet cafe in Chicago to a lifeguard station in Los Angeles and car dealership in Columbus, Ohio.
At a polling station in Fort Lauderdale, Florida, voters queued in their hundreds, many of them seeking shelter from the sun under palm trees.
In Chicago, first-time voter Galicia Malone went into labour but ensured she cast her ballot on her way to hospital.
'Work not done'
Mr Obama was the first of the two rivals to cast their ballot by voting last month in his adopted home city of Chicago, and becoming the first sitting president to vote before election day.
Mr Romney and his wife Ann voted in the Boston suburb of Belmont, Massachusetts, shortly before 09:00 local time on Tuesday. He told reporters he felt "very good".
Their running mates also voted early on Tuesday. The vice-president queued with his wife, Jill, outside a polling station in Greenville, Delaware, and urged voters to "stand in line as long as you have to".
Republican Paul Ryan cast his vote with his wife, Janna, at a public library in Janesville, Wisconsin.
Each state is given a number of electoral votes in rough proportion to its population. The candidate who wins 270 electoral votes - by prevailing in the mostly winner-take-all state contests - becomes president.
Also on Tuesday's ballot are 11 state governorships, a third of the seats in the 100-member US Senate and all 435 seats in the House of Representatives.
Republicans are expected to keep control of the House, while Democrats are tipped to do the same in the Senate.
The presidential candidates spent Monday criss-crossing the crucial battleground states including Ohio, Florida, Iowa and Virginia, making final appeals to voters. They aimed to encourage their own supporters to go to the polls while also persuading the small sliver of undecided voters to back them.
Mr Romney kept up his attack on Mr Obama's economic record, telling supporters in Virginia that "if you believe America should be on a better course, if you're tired of being tired... then I ask you to vote for real change."
At a rally in Ohio, the president acknowledged frustration with the still-lagging economy but told voters "our work is not done yet".
Legal battles feared
Legal battles feared
With voter turnout seen as vital for both candidates, campaigning continued on Tuesday, but at a slower pace.
Both men gave radio interviews, and Mr Romney hit the campaign trail again, with visits to Ohio and then Pittsburgh, Pennsylvania.
Analysts tried to determine whether Mr Romney's last-minute campaigning showed strength or desperation, while the candidate reportedly told aides he wanted to be active on election day instead of just sitting around waiting for results.
The Republican was due to return to Boston later for an election night rally.
Mr Obama, who will hold an election night rally at a convention centre in Chicago, visited a campaign office in the city to talk to volunteers. He praised his Republican rival's "spirited campaign".
With observers anticipating a close race, both sides have readied teams of lawyers for possible legal fights, especially in the critical battleground state of Ohio.
With observers anticipating a close race, both sides have readied teams of lawyers for possible legal fights, especially in the critical battleground state of Ohio.
Some analysts fear the election will not be decided on Tuesday night if the state's vote becomes mired in legal battles.
Some analysts fear the election will not be decided on Tuesday night if the state's vote becomes mired in legal battles.
On Tuesday Mr Romney is to hit the campaign trail again with events in Pittsburgh, Pennsylvania and Cleveland, Ohio, before holding an election night rally in Boston.
