Ahead of the talks, Mrs May said meeting with Mr Putin did not mean a return to "business as usual" with Russia (Picture: PA).
Theresa May condemned Vladimir Putin's "irresponsible" actions in a meeting between the two leaders dominated by exchanges about the Salisbury nerve agent attack.
The Prime Minister met the Russian president before talks in which she told him the use of the Novichok nerve agent in the Wiltshire city was a "truly despicable act".
The gathering of the two leaders took place at the G20 Summit in Japan on Friday - their first formal meeting since the Salisbury poisoning in March 2018.
The UK believes Russia's GRU military intelligence agency was behind the attack on former double agent Sergei Skripal and his daughter Yulia.
Mr Skripal and his daughter both survived but a British woman later died after coming into contact with a Novichok-contaminated perfume bottle.
Scotland Yard and the CPS believe there is sufficient evidence to charge two Russians - who go by the names of Alexander Petrov and Ruslan Boshirov - with offences including conspiracy to murder over the attack on the Skripals.
Online investigation group Belingcat said Boshirov is actually the highly decorated Colonel Anatoliy Chepiga, and Petrov is a military doctor called Alexander Mishkin.
The pair said they are innocent and Mr Putin has always insisted they are civilians, not criminals.
In an interview with Forces News, Sir Tony Breton, the former British Ambassador to Moscow, said it is important for the Prime Minister Theresa May to be tough in talks with Russia.
"The Russians, to the extent that they have cooperative ideas to offer, will be saving them up for whoever succeeds her [as Prime Minister]," said Sir Breton.
Watch: Russia 'saving co-operative ideas' for next prime minister.
More than 120 British military personnel helped with the decontamination of the city following the Novichok incident.
Speaking of the meeting between Mrs May and Mr Putin, a Downing Street spokesperson, said: "She told the president that there cannot be a normalisation of our bilateral relationship until Russia stops the irresponsible and destabilising activity that threatens the UK and its allies - including hostile interventions in other countries, disinformation and cyber attacks - which undermine Russia's standing in the world.
"The Prime Minister said that the use of a deadly nerve agent on the streets of Salisbury formed part of a wider pattern of unacceptable behaviour and was a truly despicable act that led to the death of a British citizen, Dawn Sturgess."
Downing Street said Mrs May was "clear that the UK has irrefutable evidence that Russia was behind the attack - based on painstaking investigations and co-operation with our allies.
"She said that this behaviour could never be repeated and that the UK wants to see the two individuals responsible brought to justice."
The Russian president has dismissed the Salisbury incident as "fuss about spies and counter-spies" that was "not worth serious interstate relations" and said "traitors must be punished".
The UK expelled 23 Russian diplomats that Mrs May claimed were undeclared intelligence officers following the Salisbury attack.
Other allies, including the US, followed suit.