The blast happened in the foyer at 22:35 BST on Monday at the end of a concert by US singer Ariana Grande.
Police said the lone male attacker, who died in the blast, detonated a homemade bomb.
Relatives are using social media to hunt for missing loved ones, and an emergency number, 0161 856 9400, has been set up.
The prime minister has chaired a meeting of the government’s emergency Cobra committee and is expected to travel to Manchester later.
Chief Constable Ian Hopkins said it was “the most horrific incident” the city had ever faced.
He said the “fast-moving investigation” was now working to establish whether the attacker “was acting alone or as part of a network”.
Sixty ambulances attended the incident and those wounded are now being treated at six hospitals around the city.
Eyewitnesses described seeing metal nuts and bolts among the debris, and spoke about the fear and confusion that gripped the concert-goers.
Andy Holey, who had gone to the arena 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.
“When I got up I saw bodies lying on the ground. My first thought was to go into the arena to try to find my family.”
Emma Johnson said she and her husband were at the arena to pick up her children, aged 15 and 17.
“We were stood at the top of the stairs and the glass exploded – it was near to where they were selling the merchandise,” she told BBC Radio Manchester.
“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.
“We were just trying to figure where everyone was. It was absolutely terrifying.”
Charlotte Campbell’s daughter Olivia has been missing since the concert.
“She’s only a 15-year-old girl, she’s out there on her own because her friend has been found,” she told the BBC.
“If anyone sees her contact me. Give her your phone and let her ring me. I just want her home.”
- The explosion happened shortly after Ariana Grande left the stage as concert-goers began heading to the exits
- The 23-year-old teen actress-turned-singer tweeted: “broken. from the bottom of my heart, I am so so sorry. I don’t have words”
- Home Secretary Amber Rudd said it was “a barbaric attack, deliberately targeting some of the most vulnerable”
- Flags are flying at half-mast outside Number 10 and political parties have suspended general election campaigning.
- London Mayor Sadiq Khan said the police presence in the capital would be stepped up
- World leaders have expressed solidarity with the UK, including US President Donald Trump, who called those responsible “evil losers”