The battleground of Georgia is one of several states where voting has just ended. Balloting continues across parts of the US for a few more hours.
The vote caps a long and bitter campaign amid the coronavirus pandemic.
More than 100 million people had already cast their ballots in early voting.
The US appears on course for its highest turnout in a century. The first projections are expected shortly from states which are solidly Democratic or Republican.
Control of Congress is also at stake. As well as the White House, Republicans are vying to hang on to a Senate majority.
The House of Representatives is expected to stay in Democratic hands.
At 19:00 EST (00:00 GMT) polling stations closed in Georgia, Indiana, Kentucky, South Carolina, Vermont and Virginia.
Georgia could go either way in the White House race. There have been no surprises in the other five states so far.
The BBC projects Mr Trump has won Indiana. US media project Mr Biden has won Vermont. CBS News, the BBC's US partner, says Kentucky and South Carolina are leaning Mr Trump's way, while Virginia is leaning Mr Biden's way.
But in North Carolina, where voting ends at 19:30 EST, the race was forecast to be close.
Half an hour later, voting will end in the pivotal swing states of Florida and Pennsylvania, which look like must-wins for Mr Trump if he is to be re-elected to a second term in office. Voting ends on the US West Coast at 23:00 EST.
National polls give a firm lead to Mr Biden, but it is a closer race in the states that could decide the outcome.
In the US election, voters decide state-level contests rather than an overall, single, national one.
Appearing in his home state of Delaware, Mr Biden said he was "hopeful" and highlighted the "overwhelming turnout particularly of young people, of women".
A senior Biden adviser told CBS News - the BBC's US partner - that the Biden team "feel good". Florida was in the balance, but Democratic numbers were strong in a swathe of swing states, including Wisconsin, Pennsylvania, North Carolina, Georgia and Arizona.
Speaking in Virginia, Mr Trump said he expected "a great night".
An exit poll conducted by Edison Research and just published by Reuters suggests that four out of 10 voters nationally think the handling of the coronavirus pandemic in the US is "going very badly".
A third of voters cited the economy as the issue that most concerned them, according to the poll.
Both candidates have been using election day as an opportunity to drive home the messages they have been hammering to voters for the last few weeks.
The two rivals have radically different policies on key issues. They have clashed bitterly during the campaign over how to handle the Covid-19 pandemic, and how to best handle the economy during this difficult period.
Mr Biden has accused Mr Trump of a haphazard response to the pandemic that he says has needlessly cost too many lives. But Mr Trump has downplayed the impact of Covid-19, saying the country is "rounding the corner".
Mr Trump's term has also been marked by contentious immigration policies.
Earlier on Tuesday, Mr Trump, sounding a little hoarse, spoke to Fox News by phone, saying he felt good about his chances of victory and predicting he would win "big" in key states such as Florida and Arizona.
"I think we have a really solid chance of winning," he said. Asked when he would declare victory, he added: "When there's victory. If there's victory... there's no reason to play games."
Mr Biden also visited his childhood home in Scranton in Pennsylvania - a key swing state - and addressed a crowd in the town, saying: "We've got to restore the backbone to this country. The middle class built this country - Wall Street didn't."
How will the election work?
To be elected president, a candidate must win at least 270 votes in what is called the electoral college. Each US state gets a certain number of votes partly based on its population and there are a total of 538 up for grabs.
This system explains why it is possible for a candidate to win the most votes nationally - as Hillary Clinton did in 2016 - but still lose the election.
Control of the Senate is also at stake in these elections, with the Democrats seeking to gain control of both houses of Congress and the White House for the first time since early in Barack Obama's first term.
Coronavirus has at times overshadowed the campaign, with the pandemic in the US worsening over the final weeks. The country has recorded more cases and more deaths than anywhere else in the world, and fear of infection has contributed to an unprecedented surge in early and postal voting.
There are fears that pockets of post-election violence could break out as the results come in.
A new "non-scalable" fence has been put up around the White House in Washington DC. Businesses in the nation's capital and also in New York City have been seen boarding up their premises due to concerns about unrest.
In the last hours of the campaign, Twitter and Facebook labelled a post by President Trump as "misleading", after he claimed that postal ballots in Pennsylvania could lead to rampant fraud. The social media giants also added a link to a website explaining why mail-in votes were safe.
The US Supreme Court ruled Pennsylvania could count postal ballots received three days after the election. Mr Trump and his campaign indicated they would sue to block the move.
Legal fights over ballots have also been unfolding in Minnesota, North Carolina and Texas.
When will we get a result?
It can take several days for every vote to be counted after any US presidential election, but it is usually pretty clear who the winner is by the early hours of the following morning.
However officials are already warning that we may have to wait longer - possibly days, even weeks - for the result this year because of the expected surge in postal ballots.
Different states have different rules for how - and when - to count postal ballots, meaning there will be large gaps between them in terms of reporting results. In some states it will take weeks to get complete results.
The last time the result was not clear within a few hours was in 2000, when the winner, George W Bush, was not confirmed until a Supreme Court ruling was made a month later.
The president is to host an election night party inside the White House with about 400 guests invited.
Up to 10,000 protesters are expected to gather at the renamed Black Lives Matter Plaza and a park in Washington DC, not far from the White House, according to CBS News.
Mr Biden and Ms Harris will watch the election night returns in the former vice-president's hometown of Wilmington, Delaware.
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Your Questions Answered: What questions do you have about the US election?
How does vote counting work? When will we get the final results? The US election can be confusing, especially this year. The BBC is here to help make sense of it. Please send us your questions about election day and beyond.
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