The third London Bridge attacker has been named as 22-year-old Youssef Zaghba, a Moroccan-Italian man.
Pakistan-born Khuram Butt, 27, and Rachid Redouane, 30, both from Barking were the other two attackers.
Two more victims have been been named - Australian nurse Kirsty Boden, 28, and French national Alexandre Pigeard, 27, died in Saturday night's attack.
They were among seven people killed - 48 were also injured - before the three attackers were shot dead by police.
Zaghba, Butt and Redouane drove a hired van into pedestrians on London Bridge at 21:58 BST before stabbing people in the area around Borough Market.
Armed officers killed all three within eight minutes of receiving a 999 call.
The Metropolitan Police said Butt had been subject to an investigation in 2015, but there had been no suggestion this attack was being planned.
In other developments:
- Theresa May says she expects a review will be launched by the police and security services following the attack, amid an election row over police numbers
- The Metropolitan Police said a 27-year-old man had been arrested in Barking on Tuesday in connection with the investigation
- A property in Ilford, east London, was also searched by police at about 01:30 BST, but no arrests had been made, the Met said
- A man was arrested in Limerick, Ireland, on Tuesday over the discovery of ID documents in Redouane's name. He was later released without charge.
- A second man, who is in his 30s, was arrested on Tuesday evening, with the Garda saying it was also related to Redouane. He is being held in Wexford.
- The Prince of Wales and Duchess of Cornwall visited injured members of the public at the Royal London Hospital
- NHS England said 32 people remained in hospital, with 15 in a critical condition
- A national one-minute silence was held in the UK in memory of those who were killed
- All 12 people arrested on Sunday after the attack have now been released without charge
The so-called Islamic State (IS) group has said its "fighters" carried out the attack.
An Italian police source has confirmed to the BBC that Zaghba, who lived in east London, had been placed on a watch list, which is shared with many countries including the UK.
In March 2016, Italian officers stopped Zaghba at Bologna airport and found IS-related materials on his mobile phone. He was then stopped from continuing his journey to Istanbul.
Redouane was a chef who also used the name Rachid Elkhdar and police said he claimed to be Moroccan-Libyan. He married a British woman in Dublin in 2012 and lived in Rathmines in the Irish capital.
What did police know about Khuram Butt?
Butt featured in a Channel 4 documentary last year about Islamist extremists with links to the jailed preacher Anjem Choudary called The Jihadis Next Door.
The married father-of-two, who worked for London Underground as a trainee customer services assistant for nearly six months last year, could be seen in the programme arguing with police officers in the street, after displaying a flag used by IS in a London park.
Two people in Barking, east London, had also raised concerns about Butt, BBC home affairs correspondent Dominic Casciani said.
Metropolitan Police assistant commissioner Mark Rowley said an investigation into Butt began in 2015, but "there was no intelligence to suggest that this attack was being planned and the investigation had been prioritised accordingly".
At any one time there are about 500 active counter-terrorism investigations concerning 3,000 people of interest.
Who were the victims?
So far four of those killed in the attack have been named.
Ms Boden worked as a senior staff nurse at Guy's and St Thomas' Hospital in London. Her family have described her in a statement as an "outgoing, kind and generous person".
"We are so proud of Kirsty's brave actions which demonstrate how selfless, caring and heroic she was, not only on that night, but throughout all of her life," they added.
The hospital said that Ms Boden was "an outstanding nurse and a hugely valued member of the staff team in Theatres Recovery, described by her colleagues as 'one in a million' who always went the extra mile for the patients in her care".
Canadian national Chrissy Archibald, 30, was the first victim to be named. Her family said she had died in her fiancé's arms after being struck by the attackers' speeding van.
The family of 32-year-old James McMullan, from Hackney, east London, say they believe he also died.
Mr McMullan's sister said he was believed to be among those who died, after his bank card was found on a body at the scene.
A French national was also killed in the attack, according to foreign minister Jean-Yves Le Drian, which the BBC understands to be Mr Pigeard.
Manager of the Boro Bistro, Vincent Le Berre told the Brittany news outlet Le Telegramme how his colleague was attacked in a bar near Borough Market.
"I managed to escape him, but my friend Alexandre did not have that chance," he said. "He was hit in the neck with a knife."
The Met have set up a casualty bureau on 0800 096 1233 and 020 7158 0197 for people concerned about friends or relatives.