Amid angry shouting and name calling, the two fought over the pandemic, white supremacy and the economy during the 90-minute forum in Cleveland, Ohio.
Mr Biden called the president a “clown” and told him to “shut up”. Mr Trump brought up drug use by his rival’s son.
Opinion polls suggest Mr Biden has a steady single-digit lead over Mr Trump.
But with 35 days until election day, surveys from several important states show a closer contest.
Polls also suggest one in 10 Americans have yet to make up their mind who to vote for. But analysts said Tuesday night’s debate – the first of three – probably would not make much difference.
What were the key moments?
Overall, the debate was light on policy, with little serious discussion or argument over what either candidate would do while in office.
The tenor became clear early on, as the two candidates sparred over healthcare. Frequent interruptions from Mr Trump saw Mr Biden call the president a “clown”.
As they moved on to the Supreme Court, the rancour continued, with Mr Biden refusing to answer when asked if he would try to expand the number of judges.
“Will you shut up, man?” the Democratic candidate snapped, later adding: “Keep yapping, man.”
Mr Trump, the Republican candidate seeking a second term, responded: “The people understand, Joe. 47 years [in politics], you’ve done nothing. They understand.”
In one of the most talked about exchanges of the night, the president was asked by the moderator, Fox News anchor Chris Wallace, if he was prepared to condemn white supremacists.
He initially said he would, but when asked to denounce the far-right Proud Boys group by name, he sidestepped.
Mr Trump said: “Proud Boys, stand back and stand by, but I’ll tell you what, somebody’s got to do something about antifa and the left.”
The Proud Boys, an anti-immigrant, all-male group, took to social media to celebrate. “Standing down and standing by sir,” it posted on Telegram.
Earlier, Mr Biden said: “This is a president who has used everything as a dog whistle to try to generate racist hatred, racist division.”
In other moments:
- At one point, the insults became deeply personal when Mr Trump suggested Hunter Biden, the Democrat’s son, had been thrown out of the military for cocaine use. Mr Biden said his son “had a drug problem” but had “fixed it”
- Mr Trump was asked if he would encourage his supporters to be peaceful if results of the election were unclear. “I’m encouraging my supporters to go into the polls and watch very carefully,” he responded
- When Mr Trump claimed Mr Biden would be at the behest of the left of the Democratic Party over health and environmental policy, Mr Biden responded: “I am the Democratic Party right now”
- Mr Trump defended his effort to swiftly fill a US Supreme Court seat to cement a 6-3 conservative majority on the bench, saying: “We won the election and we have the right to do it”
What did they say about coronavirus?
Mr Biden said Mr Trump had “panicked” over the coronavirus pandemic, which has killed more than 200,000 Americans.
“A lot of people died and a lot more are going to die unless he gets a lot smarter, a lot quicker,” Mr Biden said.
Mr Trump objected to Mr Biden using the word “smart.”
“You graduated either the lowest or almost the lowest in your class,” the president said. “Don’t ever use the word smart with me. Don’t ever use that word.”
Local rules required everyone in the room to wear masks, but of the president’s family members present only US First Lady Melania Trump donned a face covering during the debate.
Due to the pandemic, the forum at Case Western Reserve University had a small, socially distanced audience and the traditional opening handshake was skipped.