The blast happened at 22:35 BST on Monday following a pop concert by the US singer Ariana Grande.
PM Theresa May said her thoughts were with those affected by "what is being treated by the police as an appalling terrorist attack".
Paramedics at the scene told the BBC they had treated some of the wounded for "shrapnel-like injuries".
British Transport Police said the explosion was in the arena's foyer.
Greater Manchester Police has established an emergency telephone number in response to the attack. It is: 0161 856 9400.
The prime minister has suspended her general election campaigning and will chair a meeting of the government's emergency Cobra committee later, in response to the attack.
Mrs May said: "We are working to establish the full details" of what had happened in Manchester.
"All our thoughts are with the victims and the families of those who have been affected," Mrs May said.
BBC Home Affairs Correspondent Daniel Sandford said senior counter-terrorism officers were assembling in London and liaising with the Home Office.
Unconfirmed reports from two unnamed US officials suggested the attack was carried out by a suicide bomber.
The leader of the Labour Party, Jeremy Corbyn, tweeted: "Terrible incident in Manchester. My thoughts are with all those affected and our brilliant emergency services."
Greater Manchester metro mayor Andy Burnham said: "My heart goes out to families who have lost loved ones, my admiration to our brave emergency services. "A terrible night for our great city."
Shortly after the blast Manchester Victoria station, which is close to the concert venue, was closed and all trains cancelled.
Greater Manchester Police carried out a precautionary controlled explosion in the Cathedral Garden area of the city at about 01:32. The force confirmed it was not a suspicious item.
A number of eyewitnesses have described the confusion in the aftermath.
Andy Holey, who had gone to the arena to pick up his wife and daughter who had been at the concert, said: "As I was waiting, 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.
"When I couldn't find them, I went outside with the police and fire and looked through some of the bodies to try and find my wife and daughter.
"I managed to find them eventually and they're OK.
"It was definitely an explosion and it was some force. It happened near the box office at the entrance to the Arena."
Emma Johnson said she and her husband were at the arena to pick up her children, aged 15 and 17.
"It was definitely a bomb. It was definitely in the foyer," she told BBC Radio Manchester.
"We were stood at the top of the stairs and the glass exploded - it was near to where they were selling the merchandise.
"The whole building shook. There was a blast and then a flash of fire afterwards. There were bodies everywhere."
Michelle Sullivan, from Huddersfield, was attending the concert with her daughters, aged 12 and 15.
"It was really scary," she said. "Just as the lights have gone down we heard a really loud explosion... Everybody screamed.
"When we got out they just said 'keep on running, keep on running'."
Within an hour of reports of the incident emerging, people began offering spare rooms and beds to people stranded in the city using the hashtag #RoomForManchester.
Hundreds of tweets offering places to stay are being shared and re-tweeted thousands of times.
Pat Carney, Manchester City Council's spokesman for the city centre, said the city's thoughts were with the families of those killed and injured.
"It's a very easy target - a concert hall where young people are enjoying music," he said.
"The public are really co-operating by staying away from what is basically now a crime site.
"The world we live in, police and the council have emergency procedures that we practise all the time.
"Obviously everyone in the city is shocked, having seen how young some of these people are
"The police are treating it as a live site, we don't know if this is the end or there are other incidents in that area... we don't know at the moment."