What did the leaders discuss?
Before the summit, both sides said relations were at rock bottom.
Mr Putin hinted at a possible deal on exchanging prisoners, saying he believed compromises could be found.
On cyber-attacks, Mr Putin brushed away accusations of Russian responsibility, saying that most cyber-attacks in Russia originated from the US.
Mr Biden said he told Mr Putin that critical infrastructure, such as water or energy, must be “off-limits” to hacking or other attacks.
“I looked at him and said how would you feel if ransomware took on the pipelines from your oil fields? He said it would matter,” Mr Biden said, adding that if Russia violated these “basic norms” the US would retaliate.
The two sides differed sharply on human rights, including the right to protest.
Mr Putin dismissed US concerns about Navalny, who recently undertook a 24-day hunger strike.
He said Navalny had ignored the law and knew he would face imprisonment when he returned to Russia after having sought medical treatment in Germany. Navalny says he was poisoned with a nerve agent on Mr Putin’s orders – an accusation Mr Putin denies.
He said Russia did not want disturbances on its territory comparable to the Capitol riots or the Black Lives Matter movement.
Mr Biden dismissed Mr Putin’s comments about Black Lives Matter as “ridiculous”, and said human rights would “always be on the table”.
Asked why Russia would want to co-operate with the US, Mr Biden said it was “in a very, very difficult spot right now”.
“They are being squeezed by China. They want desperately to remain a major power,” he told reporters, shortly before leaving Geneva.
At one point during his press conference, the US president appeared to nod in response to a reporter who asked if he trusted Mr Putin. But the White House sent a tweet out soon after saying Mr Biden was “very clearly not responding to any one question, but nodding in acknowledgment to the press generally”.
When a CNN journalist asked why Mr Biden was confident Mr Putin would change his behaviour, the US president became visibly irritated, retorting: “If you don’t understand that, you’re in the wrong business.” He later apologised for being a “wise guy”.
BBC Moscow correspondent Sarah Rainsford says Mr Putin was keen to underline several times that Russia was a nuclear power – an important country, with an economy smaller than that of the US, but one that still mattered and that was why Mr Biden had come to talk to him.