The Trump campaign is challenging vote counts in the key states of Wisconsin, Georgia, Pennsylvania and Michigan.
The BBC projects Mr Biden won Michigan. US media forecast he took Wisconsin. No result has yet emerged in Pennsylvania.
Winning all three of these Rust Belt states would hand Mr Biden victory.
Mr Biden, the Democratic candidate, stopped short of declaring he had won, but said he was confident he was on course to defeat Mr Trump, his Republican rival.
Overall turnout in Tuesday’s election was projected to be the highest in 120 years at 66.9%, according to the US Election Project.
Mr Biden had the support of 70.5 million voters, the most won by any presidential candidate ever. Mr Trump has pulled in 67.2 million votes, four million more than he gained in 2016.
The bitter election race was dominated by the coronavirus pandemic, which hit a new record high of 103,000 daily cases in the US on Wednesday, according to the Covid Tracking Project.
What are the campaigns saying?
On Wednesday afternoon, Mr Biden told reporters in Wilmington, Delaware: “When the count is finished, we believe we will be the winners.”
“I will govern as an American president. The presidency itself is not a partisan institution.”
He and his running mate Kamala Harris have launched a website for the transition of power, which says that their team “will continue preparing at full speed so that the Biden-Harris Administration can hit the ground running on Day One.”
Mr Biden also said he was feeling “very good” about Pennsylvania, although President Trump’s campaign said it was “declaring victory” in the state on the count of “all legal ballots”.
Senior Trump campaign aide Jason Miller said: “By the end of this week, it will be clear to the entire nation that President Trump and Vice-President Pence will be elected for another four years.”
Can Trump still win?
Mr Biden has 243 Electoral College votes, giving him the edge in the race to accumulate the 270 needed to win the White House. Mr Trump has 214.
In the US election, voters decide state-level contests rather than a single, national one. Each US state gets a certain number of electoral college votes partly based on its population, with a total of 538 up for grabs.
If Mr Trump does lose Wisconsin (10 electoral college votes), he must win Georgia (16 votes), North Carolina (15), Pennsylvania (20) and either Arizona (11) or Nevada (6) to prevail.
Officials in Georgia said they would keep going all night until counting was finished, but as of midnight local time (05:00 GMT) they reported that there were still about 90,000 votes to count. At that time President Trump was leading by some 31,000 votes.
In Arizona, Mr Biden was leading by about 80,000 votes, with further results expected on Thursday. CBS has categorised it as a “likely” win for the Democrat. Supporters of Mr Trump gathered outside a vote counting centre in Maricopa County (which includes Phoenix, the largest city in the state), where officials vowed to “continue our job”.
An update on the count in Nevada – where the candidates were neck-and-neck – is not expected until Thursday at 17:00 GMT (09:00 local time), while in Philadelphia, counting is not expected to finish for several days.
What about the legal challenges?
The Trump campaign said the president would formally request a Wisconsin recount, citing “irregularities in several Wisconsin counties”.
Incomplete results indicate the margin between Mr Trump and Mr Biden in Wisconsin is less than one percentage point, which allows a candidate to seek a recount.
The campaign also filed a lawsuit in Michigan to stop counting there because it contended it had been denied “meaningful access” to observe the opening of ballots and the tally.
In Detroit, Michigan, police were called on Wednesday afternoon to guard the doors to a vote-counting facility as some protesters outside demanded access to monitor the process. According to the Detroit Free Press, there were already some 200 people observing the vote inside the building.
Officials were seen covering up the windows to the TCF Center, where postal ballots were being tabulated.
The Trump campaign also filed two lawsuits in Pennsylvania to halt all vote counting “until there is meaningful transparency”.
The president has a three-point lead in the Keystone State, but many thousands of votes remain to be counted.
Mr Trump is also suing Georgia to halt the vote count there. His campaign said a Republican poll observer in the southern state had witnessed 53 late absentee ballots being illegally added to a pile of votes in Chatham County.
In 2016, Mr Trump’s wins in Wisconsin, Michigan and Pennsylvania sealed his victory and took him to the White House.
In the early hours of Wednesday, the president claimed he had won the election despite many uncounted votes remaining. The speech brought strong criticism from both Democrats and Republicans.
Mr Trump also said he was prepared to take the election to the Supreme Court, and his campaign is asking Republican donors to help fund legal challenges. Such challenges start at a state level but could eventually lead to the top court in the US.
Republican National Committee Chairwoman Ronna McDaniel said: “The fight’s not over. We’re in it.”
Mr Biden’s running mate, Kamala Harris, tweeted asking supporters to contribute $5 to help pay for litigation that could “stretch on for weeks”.
Biden campaign senior legal adviser Bob Bauer said there were no grounds for Mr Trump to invalidate lawful ballots.
What were the other key results?
Mr Biden’s hopes of a decisive early victory on election night were thwarted as Mr Trump defied pollsters’ predictions by over-performing in key battlegrounds.
The president held several important states, including Texas, Ohio and Iowa.
Source: BBC
