The night opened with predictable victories for each candidate.
Mr Trump took states including Louisiana and North Dakota and Mr Biden’s haul included Colorado and Virginia – two former battlegrounds that have become Democratic strongholds.
Bu it is too early to call the results in the 2020 battleground states of Florida, Georgia and Pennsylvania.
National polls are giving a firm lead to Democratic challenger Mr Biden, but in the states that could make the difference the race is far closer.
Mr Trump began Election Day on an upbeat note, predicting that he would do even better than in 2016.
He left open the possibility of addressing the nation Tuesday, even if a winner is not yet determined. Former vice president Mr Biden promised a speech, too.
“Winning is easy,” Mr Trump told reporters. “Losing is never easy, not for me it’s not.”
The Democratic nominee kept his eyes on the critical state of Pennsylvania, taking his final pitch to voters in his birthplace of Scranton and the Democratic stronghold of Philadelphia.
In battlegrounds, including Florida, Iowa, Georgia, Michigan and Pennsylvania, some voters showed up to their polling places before dawn to beat the crowds.
The record-setting early vote drew unsupported allegations of fraud from Mr Trump, who had refused to guarantee he would honour the election’s result.
Mr Biden visited his childhood home and church in his native Scranton on Tuesday as part of a get-out-the-vote effort before awaiting election results in his hometown of Wilmington, Delaware.
His running mate Senator Kamala Harris was visiting Detroit in battleground Michigan.
Mr Biden and his wife Jill started the day with a stop at St. Joseph’s on the Brandywine in Wilmington, Delaware, with two of his grandchildren in tow.
Then they walked to his late son Beau Biden’s grave, in the church cemetery.
Beau, a former Delaware attorney general, died of brain cancer in 2015 and had encouraged the former vice president to make another White House run.
Mr Trump called into “Fox & Friends,” where he predicted he will win by a larger electoral margin than he did in 2016, when he tallied 306 electoral college votes compared to Democrat Hillary Clinton’s 232.
He has invited hundreds of supporters to an election night party in the East Room of the White House.
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