The brothers, who say they are innocent, are accused of having formed an organised criminal network in early 2021 in Romania, as well as in the US and Britain
At least 48 people were injured in the attack, the third to hit Britain in less than three months and occurring days ahead of a snap parliamentary election on Thursday.
Police shot dead the three male assailants in the Borough Market area near the bridge within eight minutes of receiving the first emergency call shortly after 10 pm local time.
"We believe we are experiencing a new trend in the threat we face as terrorism breeds terrorism," Prime Minister Theresa May said in a televised statement on Sunday in front of her Downing Street office, where flags flew at half-mast.
"Perpetrators are inspired to attack not only on the basis of carefully constructed plots ... and not even as lone attackers radicalised online, but by copying one another and often using the crudest of means of attack."
She said the series of attacks represented a perversion of Islam and that Britain's counter-terrorism strategy needed to be reviewed, adding: "It is time to say enough is enough."
- Reuters Top News (@Reuters) June 3, 2017
London Bridge is a major transport hub and nearby Borough Market is a fashionable warren of alleyways packed with bars and restaurants that is always bustling on a Saturday night.
The area remained cordoned off and patrolled by armed police and counter-terrorism officers on Sunday, with train stations closed. Forensic investigators could be seen working on the bridge, where buses and taxis stood abandoned.
There was no immediate claim of responsibility for the latest attack to hit Britain and Europe.
Less than two weeks ago, a suicide bomber killed 22 people including children at a concert by US singer Ariana Grande in Manchester in northern England. In March, in a attack similar to Saturday's, a man killed five people after driving into pedestrians on Westminster Bridge in central London.
- Reuters Top News (@Reuters) June 4, 2017
The three attackers on Saturday night were wearing what looked like explosive vests that were later found to have been fake. May said the assailants' aim had been to sow panic. The BBC showed a photograph of two possible attackers shot by police, one of whom had canisters strapped to his body.
Home Secretary (interior minister) Amber Rudd said it did not appear that there was a link between the attackers and the Manchester bombing.
Militants have carried out attacks in Berlin, Nice, Brussels and Paris over the past two years.
The brothers, who say they are innocent, are accused of having formed an organised criminal network in early 2021 in Romania, as well as in the US and Britain
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The charges against them include planning terrorist acts, fundraising for the Organisation, and concealing the source and destination of those funds