Voting precincts in the eastern United States have begun to open their doors to citizens who will cast votes for President Barack Obama or Republican challenger Mitt Romney in the hotly contested 2012 presidential election.

Millions of voters across the country have already cast ballots under early-voting rules. But the vast majority of the electorate will head to polling places Tuesday in schools, firehouses, churches and elsewhere.

President Obama and Mr. Romney dashed across several key battleground states Monday in a final effort to sway any remaining undecided voters.

Mr. Obama made campaign stops in Wisconsin and Ohio, before holding a final rally in Iowa, the state that gave him his first primary victory in his historic 2008 White House campaign. The Democratic incumbent boasted of his accomplishments during his presidency, including the bailout of the U.S. auto industry and the killing of Osama bin Laden, but said he needed another term to complete his agenda.

Mr. Romney held a rousing late-night rally in New Hampshire, where he launched his campaign more than a year ago, after events in Florida, Virginia and Ohio. The former Massachusetts governor his record as both a successful businessman and politician shows he, not Mr. Obama, would bring about real change for the nation.

Mr. Romney votes in his hometown of Boston Tuesday, and has scheduled two last-minute Election Day events in Ohio and Pennsylvania. The president and his wife, Michelle, will spend Tuesday in their home in Chicago.

Voters in the small New Hampshire towns of Dixville Notch and Hart's Location cast their ballots at midnight Tuesday, keeping with tradition in being the first locations in the nation to vote on Election Day. Mr. Obama and Mr. Romney tied at five votes each in Dixville Notch. In Hart's Location, the president won 23 votes while Mr. Romney finished with nine.

A wide collection of polls shows the two candidates in a very close race nationally. But state-by-state polls show Mr. Obama with steady, but narrow leads in most of the closely contested states likely to determine the outcome.

U.S. political analysts say a handful of the country's 50 states will decide Tuesday's election, with the remainder leaning toward or firmly in the grasp of either the president or Mr. Romney. U.S. presidential elections are not decided by the national popular vote, but rather by an electoral college system in which the importance of each state on the outcome is roughly equivalent to its population.

Along with the race for president, all 435 seats in the U.S. House of Representatives and 33 of the 100 Senate seats are being contested in Tuesday's election. Analysts generally say Republicans will continue to hold their majority in the House, while the president's Democratic party could maintain their slim majority in the Senate.