At least 133 people were killed and more than 140 injured when the attackers stormed the building, firing indiscriminately and setting it alight. Authorities said 11 people were detained in total, and the four gunmen were caught while heading to Ukraine.
The Islamic State (IS) group says it was behind the attack.
The group later released highly graphic footage from the attack. The video, which has been verified by the BBC as genuine, shows one of the gunmen opening fire on several people. The BBC will not be broadcasting this video.
In a televised address, Mr Putin condemned the massacre – the deadliest in Russia for nearly 20 years – as a “barbaric terrorist act” and repeated earlier suggestions by Russian security services that the attackers had tried to escape to Ukraine. Kyiv dismissed the claim that it was some way involved in the attack as “absurd”.
“To suggest the suspects were heading to Ukraine, would suggest they were stupid or suicidal,” Andriy Yusov, a spokesperson for the Ukrainian military intelligence directorate, told the BBC.
The US National Security Council said it warned Russia of a potential attack on “large gatherings”, including concerts, in Moscow earlier this month. The Kremlin at the time dismissed that as “propaganda” and an attempt by Washington to meddle in the recent Russian election.
On Saturday, the White House said it condemned the “heinous” attack and described Islamic State as “a common terrorist enemy that must be defeated everywhere”.
- Bullets and panic – the Moscow concert that became a massacre
- What we know about the attack on a Moscow concert hall
Reports of a serious attack inside the Crocus City Hall, in Moscow’s Krasnogorsk suburb, began arriving at around 20:00 local time (17:00 GMT) on Friday.
Up to 6,200 people were gathered there for a Friday night rock concert by veteran band Picnic when the shooting started.
One video posted online showed several men striding across the concourse, where they opened fire at members of the public, before rearming and entering the hall.
Crowds of people were filmed screaming and running in panic as the men burst in. Others were seen taking cover behind their seats as the men fired into the auditorium.