Mr Trump, a Republican, claimed to have won and vowed to launch a Supreme Court challenge, baselessly alleging fraud.
Earlier Mr Biden, a Democrat, said he was "on track" to victory.
Millions of votes remain uncounted and no candidate can credibly claim victory as yet. There is no evidence of fraud.
More than 100 million people cast their ballots in early voting before election day on Tuesday - setting US on course for its highest turnout in a century.
With the nation on edge, the final result may not be known for days.
What are the presidential results so far?
The president is projected to have held the must-win state of Florida - a major boost to his re-election bid.
But Mr Biden could snatch Arizona, a once reliably conservative state. Fox News and the Associated Press have projected Mr Biden will win that state and CBS News, the BBC's US partner, said it was leaning the Democrat's way.
A loss for Mr Trump in that once reliably Republican state would be a potentially serious setback.
Other key states such as Georgia, Pennsylvania, Wisconsin, Michigan and North Carolina are toss-ups.
The BBC projects Mr Trump will win another conservative sunbelt state, Texas, where the Biden campaign had dreamed of an upset victory.
The Rust Belt battlegrounds of Pennsylvania, Michigan and Wisconsin - which propelled Mr Trump to the White House four years ago - still look as though they could tip either way.
Pennsylvania, along with Florida, is considered a must-win for Mr Trump if he is to stave off defeat.
Cliffhanger counts are also under way in two more critical swing states on the East Coast, Georgia and North Carolina.
No surprises have emerged yet in the other states.
Mr Trump will keep hold of Ohio and Missouri, known as bellwether states because they have so often predicted the eventual winner, according to the BBC's projection.
The BBC also projects Mr Trump will win Florida, Iowa, Montana, Alabama, Mississippi, Idaho, Wyoming, South Carolina, Kansas, Nebraska, Utah, Louisiana, Indiana, North Dakota, South Dakota, Kentucky, Tennessee, Oklahoma, Arkansas and West Virginia, all as expected.
Mr Trump is also projected by the BBC to win Nebraska, though Mr Biden picked up one vote there in the electoral college, which could turn out to be crucial.
Mr Biden will retain his home state of Delaware and the battleground state of Minnesota for Democrats, along with California, Virginia, New York, Rhode Island, Oregon, Washington, Illinois, New Hampshire, Connecticut, New Mexico, Colorado, Vermont, Maryland, Massachusetts, New Jersey and Washington DC, according to BBC projections.
CBS projects Nevada, Maine and Hawaii were also trending towards Mr Biden.
National opinion polls before election day 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.
Projections of state wins 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.
Control of Congress is also at stake. As well as the White House, Republicans are vying to hang on to a Senate majority.
Republicans have lost a Senate seat in Colorado, but gained one in Alabama.
Mitch McConnell, the Republican Senate leader, and Trump ally Lindsey Graham have both been re-elected.
The House of Representatives is expected to stay in Democratic hands.
An unlikely battleground
Conservative Nebraska would seem to be an unlikely electoral battleground. There is a scenario, however, where the state - or at least part of it - could decide who becomes the next US president.
Donald Trump is projected to win here - but in a surprise, Joe Biden was able to pick up one of its five electoral college votes.
This is because Nebraska is one of two states that allots some of their electoral college votes to the winners of individual congressional districts, in addition to the victor in the overall popular vote.
Mr Biden is projected to carry one of these districts, the second, which includes the largest city in the state, Omaha.
If the presidential race is close - if, for instance, Mr Biden flips Arizona, Wisconsin and Michigan from 2016 but loses Pennsylvania - having won Omaha could be the key in determining whether the Democrat becomes president.
The year 2020, full of surprises and turmoil, may have a few tricks left up its sleeve still.
What are the candidates saying?
Mr Trump is hosting an election night gathering inside the presidential mansion with about 100 guests.
In a speech at about 02:30 local time (07:30 GMT) he said: "We were getting ready to win this election. Frankly, we did win this election."
He went on to allege "major fraud on our nation" without providing evidence. "So we'll be going to the US Supreme Court."
Millions of postal ballots have still to be counted and there is no evidence of fraud.
Earlier, at about 01:00 local time, Mr Biden predicted in his hometown of Wilmington, Delaware, that he would be victorious.
The Democrat said: "We feel good about where we are, we really do. I am here to tell you tonight we believe we're on track to win this election."
He added: "We're going to have to be patient until the hard work of tallying votes is finished and it ain't over until every vote, every ballot is counted."
As he spoke, Mr Trump tweeted from the White House: "We are up BIG, but they are trying to STEAL the Election. Twitter labelled the post as potentially "misleading about an election or other civic process".
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 boarded up their premises ahead of election day due to fears of unrest.
What does the exit poll data show?
An exit poll conducted by Edison Research and 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.
Exit poll data also suggest Mr Biden had the edge with women voters by 57% to 42%, with black voters (87% to 11%), with under-29 year olds (64% to 33%) and among voters with or without a college degree.
Mr Trump appeared to hold the advantage with over 65 year olds (51% to 48%).
How does 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.
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.
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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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