With more than 90% of votes counted in Florida, Mr Trump, the Republican incumbent, was three points ahead of Mr Biden, the Democratic challenger.
It is too early to project who won in several other key states - Georgia, Pennsylvania, Ohio and North Carolina.
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.
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.
What are the results so far?
After polling stations closed across the US East Coast, Pennsylvania, Georgia, North Carolina and Ohio looked as though they could go either way in the White House race.
Both Florida and Pennsylvania are considered must-wins for Mr Trump if he is to be re-elected to a second term in office.
It is also too early to project four other crucial battlegrounds - Arizona, Michigan, Texas and Wisconsin.
No surprises have emerged yet in the other states.
The BBC projects Mr Trump will hold on to Indiana, South Dakota, Kentucky, Tennessee, Oklahoma, Arkansas and West Virginia, all as expected.
The BBC also projects Mr Biden will keep his home state of Delaware, along with New York, Vermont, Maryland, Massachusetts, New Jersey and Washington DC in his column.
CBS News, the BBC's US partner, projects South Carolina, Alabama, North Dakota, Nebraska, Kansas, Louisiana, and Wyoming are leaning Mr Trump's way.
CBS also projects Minnesota, New Mexico, Maine, Colorado, Connecticut, Illinois, Rhode Island and New Hampshire are trending towards Mr Biden.
Mr Trump narrowly lost Minnesota in 2016 and his campaign is hoping to pick it up this time.
Voting ends on the US West Coast at 23:00 EST (04:00 GMT on Wednesday).
National opinion polls gave a firm lead to Mr Biden, but pointed to a closer race in the handful of states that are likely to decide the outcome.
Mr Trump, who is watching the returns from the White House, is expected to address the nation later on Tuesday evening.
Mr Biden was at his home with family in Wilmington, Delaware, with few aides around, according to CBS.
Projections are based on a mixture of exit poll data and, in most cases, actual votes counted - and are only made where there is a high degree of certainty.
In the US election, voters decide state-level contests rather than an overall, single, national one.
The night is just getting started
Polls in the eastern half of the US are closing fast now. While Florida appears to be trending toward Donald Trump - on the back of better-than-expected performance among Hispanics in South Florida - attention will shift towards other battleground states.
Joe Biden has taken an early lead in Ohio, although the counties that have been reporting are Democratic ones. However, if he can do better than Hillary Clinton, who lost the state, it could give clues as to how he'll perform in the nearby states that were much closer - Pennsylvania, Michigan and Wisconsin, in particular.
Georgia continues to look like a tossup, as does North Carolina. Both are pretty much must-wins for Trump. And Texas looms on the horizon. While Florida is the big electoral prize that has been a fixation for Democrats for decades, if Texas moves to the left, it could herald an even more earth-shattering electoral realignment.
Appearing earlier on election day in Delaware, Mr Biden said he was "hopeful" and highlighted the "overwhelming turnout particularly of young people, of women".
A senior Biden adviser told CBS 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, said the aide.
Speaking in Virginia earlier, 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.
Who decides which candidate wins a state?
The final election results don't get certified for days or even weeks, so it falls to US media organisations to predict, or project, the winner in each state much sooner. Teams of election experts and statisticians analyse a mixture of information such as exit poll data - interviews at polling stations and phone calls with early voters - and actual votes counted. In a state that always votes for one party, the results are sometimes projected as soon as voting ends, based on exit polls. In a closer contest, however, the data will draw heavily on the actual count.
This year the BBC gets its data from polling firm Edison Research who do the field work for the exit polls and work with US networks, ABC, CBS, CNN and NBC. Record levels of early voting have complicated this count, so there is no race to be first. If the BBC and its partners don't believe there is enough data to project a winner, they won't - even if others are doing so.
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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