Candidates in this odd-year election headed to the polls Tuesday morning, presumably to cast ballots for themselves, and then got underway gladhanding voters at poll stations across the country.

In Virginia, Democrat millionaire businessman Mark Warner cast his vote in Richmond at 7:30.  Republican Attorney General Mark Earley was escorting his parents to the polls before casting his vote.

New York's Democrat Mark Green and Republican Michael Bloomberg started their meet-and-greets just after the polls opened at 6 a.m.

Three key races are taking place in states touched by terrorism.  In New York City, the nasty mayoral race to succeed Rudolph Giuliani has candidates arguing they are better able to handle Ground Zero recovery. In New Jersey, a gubernatorial contest favors a Democrat in a Republican-held state where investigators are also focusing on the anthrax probe. And Virginia, home of the scarred Pentagon where the war on terror is run, is also set to vote on a new governor.

In all, Election Day will decide the mayors of 362 cities, including Atlanta, Boston, Cleveland, Houston, Miami, Minneapolis, and Seattle.

Miami has 10 candidates, Houston has six running for mayor. But no race has had the attention, spending or stakes that have gone with the race to succeed New York's Giuliani.

Democrat Mark Green, who has spent eight years as public advocate and become a regular on the election circuit, is running against media mogul Republican Michael Bloomberg. Bloomberg has spent more than $40 million of his own money on his campaign.

Both insist that they are better qualified to lead the city's recovery from the Sept. 11 terrorist attack. Each accused the other of mudslinging and racially divisive tactics.

Green held a big lead in the polls until Bloomberg finally gained Giuliani's endorsement last week. On Monday, a Quinnipiac University poll had the two running at 42-42 percent, with 15 percent of the vote undecided.

Two weeks ago, Green was beating Bloomberg 51-35 but that was before Giuliani taped a commercial endorsing Bloomberg that has saturated the airwaves.

Giuliani has received near-universal support for his handling of the Sept. 11 attacks that brought down the World Trade Center and killed more than 4,000 people.  Shortly following the attacks, the state Legislature briefly dawdled with the idea of extending the mayor's term or lifting term limits.

Bloomberg also ran an ad that quotes Green as saying he could have handled the World Trade Center tragedy as well as or better than Giuliani. Then the commercial asks, "Really?"

Green has countered by noting that Bloomberg once belonged to whites-only clubs and was the target of a sexual harassment suit by a former employee in 1997. He also has the advantage of running in a town that is five-to-one registered Democrat, though turnout, notoriously unpredictable, continues to be the unknown for all Tuesday's races.

In the nation's only gubernatorial races, polls gave sizable leads to moderate Democrats Jim McGreevey in New Jersey and Mark Warner in Virginia. Both states have Republican governors who are not seeking re-election.

McGreevey, a suburban mayor, skipped campaigning for a third straight day Monday to be with his pregnant wife, who is hospitalized. Wife Dina is seven months pregnant and suffering from mid-term labor, which doctors are trying to arrest. 

McGreevey's campaign said he would make only two public appearances Tuesday -- to vote late in the day and to declare victory or concede defeat. 

Former Jersey City mayor Republican Bret Schundler cast his vote with his family Tuesday morning at Jersey City Library.

Shundler was running behind McGreevey 39-48 in Monday's Quinnipiac poll with 12 percent undecided. A conservative, Schundler won a stunning upset primary victory for the GOP nomination by blasting the state's more moderate Republican establishment. That set him up for criticism in the general election from McGreevey, who painted him as out of touch with New Jersey voters.

In Virginia, Warner, who made a fortune with high-tech investments, has benefited from Republican infighting over the state's fiscal troubles. GOP opponent Earley has tried to capitalize on Warner's lack of political experience. Warner has run for Senate against Sen. John Warner and was chairman of the state Democratic Party but has never held elected office.

The race began on a nasty note but softened after the Sept. 11 attacks. Warner, however, has outspent Earley every step of the way, and has run on traditionally Republican themes, including tax cuts and gun rights, an influential issue in the state that houses the National Rifle Association.

Earley is also suffering from structural problems in Virginia's GOP organization. The party had a spectacular run in recent years, taking the governor, lieutenant governor, House of Delegates, and state Senate, but did not plan ahead for the 2002 campaign in terms of message, money or organization.

The Republican National Committee, led by Va. Gov. Jim Gilmore, did give Gilmore's attorney general $3 million but it does not compare to funding Warner has drawn from his personal fortune.

Seeking to avoid overt partisanship at a time of national crisis, President Bush decided not to campaign personally for Schundler or Earley, though he extended long-distance support by signing letters endorsing the candidates.

"The president just makes those judgments on the basis of what he thinks is right," said White House spokesman Ari Fleischer, who said Tuesday's results would reflect local issues rather than any verdict on Bush's anti-terrorism strategy.

Heading into Election Day, there were 29 Republican governors, 19 Democrats and two independents.

In some cities with mayoral elections — such as Boston and Pittsburgh — incumbents were expected to breeze to re-election.

Other incumbents faced stiff challenges, and others are focused on sensitive racial issues, and involve black and white candidates competing against each other:

— In Cincinnati, white incumbent Charlie Luken was challenged by Courtis Fuller, a black political newcomer. Both are registered Democrats; both were news anchors at the same local TV station. Cincinnati is 43 percent black in a city of 331,000.

Fuller has criticized Luken for imposing a curfew in September after a white officer was acquitted in the fatal shooting of a black man. The death set off rioting in April.

— In Minneapolis, community activist R.T. Rybak, who is white, was favored over incumbent Sharon Sayles Belton, the city's first black and first female mayor. Both are Democrats; Rybak is considered more liberal.

— Houston's first black mayor, Lee Brown, is seeking a third term. Brown served as former President Clinton's drug czar before Gen. Barry McCaffrey and was police chief in Atlanta, Houston and New York. His main challengers were City Councilmen Chris Bell, who is white, and Cuban-born Orlando Sanchez. Brown and Bell are Democrats; Sanchez, who would be Houston's first Hispanic mayor, is Republican.

Atlanta, Cleveland, Detroit, and Seattle will all get new mayors after incumbents depart. Outgoing Atlanta Mayor Bill Campbell's administration has been entangled in corruption scandals; Seattle Mayor Paul Schell lost in a primary, at least in part because of the city's struggles to cope with rioting during a World Trade Organization meeting in late 1999.

Among the many local and statewide ballot initiatives to be decided Tuesday are two in Washington state:

One would impose the highest tobacco taxes in the nation, boosting the price of a cigarette pack to nearly $5. The other would prohibit local governments from raising property taxes by more than 1 percent each year unless voters authorize a larger increase.

Opponents of the property-tax initiative noted that the inflation rate often is more than 1 percent, and warned that local officials would be forced into making damaging budget cuts if the measure passed.

Fox News' Carl Cameron and the Associated Press contributed to this report.