In a statement in Downing Street on Tuesday, the prime minister said it had been a "callous terrorist attack" that targeted "defenceless young people".
She said the security services believe they know the attacker's identity but are not yet able to confirm it.
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.
The BBC understands the perpetrator was British or from UK.
Greater Manchester Police Chief Constable Ian Hopkins said it was "the most horrific incident" the city had ever faced, and 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.
- 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.
- Theresa May chaired a meeting of the government's emergency Cobra committee and is now visiting Manchester
- Sporting events and venues in England are conducting major security reviews and London Mayor Sadiq Khan said the police presence on the capital's streets would be stepped up
- 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 has extended her "deepest sympathy" to all those affected, while Pope Francis offered "heartfelt solidarity" with the victims and their families
- Take That have cancelled a concert in Liverpool on Tuesday night "out of respect"