Salman Abedi has been named by police as the suspected suicide bomber who killed 22 people and injured 59 at Manchester Arena on Monday night.
The 22-year-old was Manchester born and from a family of Libyan origin, the BBC understands.
So far three victims have been named – Saffie Rose Roussos, eight, Georgina Callander and John Atkinson, 28.
Greater Manchester Police said the priority was to establish whether Abedi had worked alone or not.
A vigil is being held in front of the town hall in Manchester’s Albert Square.
Abedi is thought to have blown himself up in the arena’s foyer shortly after 22:30 BST on Monday, as fans were beginning to leave a concert by US singer Ariana Grande.
Greater Manchester Police Chief Constable Ian Hopkins passed on “heartfelt sympathies to all the innocent people caught up in last night’s despicable act”, adding that specially-trained family liaison officers were supporting families.
Eight-year-old Saffie Rose Roussos was a pupil at Tarleton Primary School, in Lancashire.
Her head teacher, Chris Upton, said she had been “simply a beautiful little girl in every aspect of the word” and was “loved by everyone”.
John Atkinson was from Bury in Greater Manchester.
Student Georgina Callander, believed to have been 18, has also been named as among the dead.
She had been studying health and social care at Runshaw College in Leyland, Lancashire.
The wounded are being treated at eight hospitals around the city, with 12 children under the age of 16 among them.
Several people are still missing, including teenagers Laura MacIntyre and Eilidh MacLeod, from Barra in the Outer Hebrides, 15-year-old Olivia Campbell, Chloe Rutherford, 17, and Liam Curry, 19.
Scotland’s First Minister Nicola Sturgeon said police were in contact with Laura and Eilidh’s families, adding: “It is hard for any of us to imagine the anguish that their families are going through right now.
“They are in our thoughts.”
In a statement in Downing Street on Tuesday, the prime minister said the bombing had been a “callous terrorist attack” that targeted “defenceless young people”.
Number 10 later said Mrs May – who is now in Manchester – had been updated “through [Monday] night” and had phoned Labour leader Jeremy Corbyn at 04:00 BST to brief him.
It is the worst terrorist attack in the UK since the 7 July bombings in 2005, in which 52 people were killed by four suicide bombers.
So-called Islamic State has said – via IS channels on the messaging app Telegram – it was behind the Manchester attack, but this has not been verified.
In other developments:
- Relatives are using social media to hunt for missing loved ones, and an emergency number – 0800 096 0095 – has been set up
- Flags are flying at half mast outside Number 10 and political parties have suspended general election campaigning until further notice
- Theresa May chaired a meeting of the government’s emergency Cobra committee and is now in Manchester where she has visited the children’s hospital and signed the book of condolence at the Town Hall
- Extra armed officers will be deployed to Wembley and Twickenham on Saturday, while security at all upcoming events and venues in England are under review. The Met Police has also increased the numbers of officers on duty across the capital
- World leaders have expressed solidarity with the UK, including US President Donald Trump, who called those behind the attack “evil losers”
- Exam boards are telling schools directly affected by the attack that they can re-arrange GCSE and A-level exams in the wake of the attack
- Police have established a help centre at Manchester City’s Etihad Stadium, access Gate 11, for anyone who needs assistance in tracing loved ones
- The Queen extended her “deepest sympathy” to all those affected and other senior royals have said they are “shocked and saddened”; Pope Francis offered “heartfelt solidarity” with the victims and their families
- Take That are among a number a performers who have cancelled concerts “out of respect”, including for the rest of the week at Manchester Arena
Witnesses at the arena described seeing metal nuts and bolts among the debris of Monday’s bomb, and spoke about the fear and confusion that gripped concert-goers.
Andy Holey, who had gone to pick up his wife and daughter, said: “An explosion went off and it threw me about 30ft from one set of doors to the other set of doors.”
Emma Johnson, who went to pick up her children, aged 15 and 17, said: “The whole building shook. There was a blast and then a flash of fire afterwards. There were bodies everywhere.”
Teenager Abigail Walker, who was at the concert, told the BBC: “I had to make sure I had my sister. I grabbed hold of her and pulled hard. Everyone was running and crying.
“It was absolutely terrifying.”
The explosion happened shortly after US singer Ariana Grande had left the stage and the 23-year-old actress-turned-singer, tweeted: “broken. from the bottom of my heart, i am so so sorry. i don’t have words”.
Abedi, who had at least three siblings, had lived at several addresses in Manchester, including a property at Elsmore Road, Fallowfield, which was earlier raided by police.
Armed police have also arrested a 23-year-old man in Chorlton, south Manchester, in connection with the attack.
Mr Hopkins said searches at two addresses had been carried out, including the one in Fallowfield, where a controlled explosion had been used to gain “safe” access.
He said Abedi had not been formally identified and so would not comment further.