Twenty-two people, including an eight-year-old girl, were killed and 59 injured when a suicide bomber attacked concertgoers at Manchester Arena.
A man set off a bomb in the foyer at 22:33 BST on Monday, at the end of a concert by Ariana Grande. Theresa May called the attack "sickening".
Armed police have arrested a 23-year-old man in Chorlton, south Manchester, in connection with the attack.
The Queen held a minute's silence at Buckingham Palace at 16:00 BST.
Three victims have been named so far, the latest being 28-year-old John Atkinson from Bury, Greater Manchester.
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".
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".
She said the security services believed they knew the attacker's identity but were not yet able to confirm it.
Number 10 later said Mrs May 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.
The BBC understands the perpetrator was British or from the UK.
Greater Manchester Police Chief Constable Ian Hopkins said the "fast-moving investigation" was now working to establish whether the attacker "was acting alone or as part of a network".
Officers have carried out raids at two properties, one in Whalley Range and one in Fallowfield, where a controlled explosion was carried out.
In its latest statement the force said the area around the arena remained cordoned off but said it wanted to "remind people that Manchester will not be defeated - the city is open for business."
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".
Analysis: Gordon Corera, BBC security correspondent
The UK threat level has been judged to be severe for nearly three years - which means an attack is considered highly likely.
But in recent months the tempo of counter terrorist activity has been increasing with - on average - an arrest every day.
After the attack in Westminster by Khalid Masood in March, police and security officials have been warning that further attacks were almost inevitable.
But they also believed that those were more likely to be low-tech involving knives or vehicles. The fact that the Manchester attack involved explosives will worry them.
It may not have been at the level of complexity seen in Paris in 2015, when multiple attackers sent from Syria used guns and suicide belts, but it will still have required planning.
The blast happened close to the entrance to Victoria railway and tram station. The station has been closed and all trains cancelled.
Greater Manchester Mayor Andy Burnham said the city would "pull together", adding: "That's what we are. That's what we do. They won't win."
The Manchester Arena is the city's largest indoor venue with a concert capacity of about 21,000.
A spokesman for the arena called the event "a senseless tragedy", but praised the emergency services for their response.
Police are encouraging anyone with footage from the scene to upload it at ukpoliceimageappeal.co.uk or ukpoliceimageappeal.com. Other information can be reported to the anti-terrorist hotline on 0800 789 321.