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 born in Manchester to parents of Libyan descent, the BBC understands.
Three of his 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.
The former Salford University student 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.
A vigil for the dead and injured was held in front of the town hall in Manchester's Albert Square on Tuesday evening.
Earlier, 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 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, Greater Manchester.
Student Georgina Callander, believed to have been 18, had been studying health and social care at Runshaw College in Leyland, Lancashire.
The wounded, who include 12 children aged under 16, are being treated at eight hospitals across Manchester.
Several people are still missing, including Eilidh MacLeod, 14, from Barra in the Outer Hebrides, 15-year-old Olivia Campbell, Chloe Rutherford, 17, and Liam Curry, 19.
Eilidh's friend, Laura MacIntyre, 15 - who was also reported as missing - was later identified as one of the seriously injured in a Manchester hospital.
Thousands of people turned out for the vigil in Manchester and to hold a minute's silence to remember those who died.
Home Secretary Amber Rudd, Labour leader Jeremy Corbyn and Commons Speaker John Bercow stood on stage alongside Manchester Mayor Andy Burnham and Greater Manchester Police Chief Constable Ian Hopkins.
Vigils have been held elsewhere, including in Birmingham, where the event was interrupted after a man apparently armed with a large knife and a baseball bat was detained nearby.
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
- The prime minister and the Labour leader both signed the book of condolence at Manchester's Town Hall
- Extra armed officers will be deployed to Wembley and Twickenham at the weekend, while security at all upcoming events and venues in England is under review.
- 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; 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". They have cancelled four shows this week, including three in Manchester
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 visited a children's hospital in Manchester, had been updated throughout Monday night and had phoned Labour leader Jeremy Corbyn at 04:00 BST to brief him.
The arena bombing 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.
The director general of the security service MI5, Andrew Parker, condemned the terrorist attack as "disgusting" and said the security service remained "relentlessly focused, in numerous current operations, on doing all we can to combat the scourge of terrorism and keep the country safe".
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 was waiting for 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 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 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 raided by police on Tuesday.
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.