The cause is unknown but PM Theresa May said her thoughts were with those affected by “what is being treated by the police as an appalling terrorist attack”.
British Transport Police said the explosion was in the arena’s foyer.
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 BBC’s Tim Ashburner, who is at the scene, spoke to some volunteer paramedics who treated the injured for “shrapnel-like injuries”.
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.
Prime Minister Theresa May has said her thoughts are with the victims and families of those affected in “what is being treated by the police as an appalling terrorist attack”.
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.”
‘Screaming and running’
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 their daughters, 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.”
At the scene: Tom Mullen BBC News
The police activity around the arena has been huge. There are blue flashing lights and cordons seemingly on almost every street corner.
A wide area around the venue itself has been completely taped off, and the crime scene appears to be widening, with police pushing people further and further back.
I’ve spoken to people who are shaken, scared and often tearful. One thing that’s apparent is there are many, many young people, some of them with parents or guardians. One mother told me her priority was simply to get her daughters home.
Other people have been more candid and have described seeing people covered in blood, or being treated by paramedics. There’s still a huge sense of confusion and people are constantly searching for information while letting their families know they’re safe.
Josh Elliott, speaking to BBC Radio 5 Live, said he was shocked by news of the fatalities.
“A bang went off and everyone stopped and screamed… we basically hit the deck,” he said
“It was bedlam… it was horrific.
“We got up when we thought it was safe and got out as quickly as possible.
“People were just crying and in tears… police cars were everywhere.
“We just wanted to get out as quickly as possible because we didn’t know what was going on.”
The background – Ian Youngs, BBC News entertainment reporter
Manchester Arena, formerly known as the MEN Arena, is the biggest indoor venue in the city with a capacity of around 18,000 for concerts.
The arena foyer connects with Victoria train and tram station, a major hub on the northern edge of the city centre.
The arena regularly hosts concerts by major stars like Ariana Grande – a 23-year-old American TV teen actress-turned-pop star.
She’s a big draw for young fans, with hits including Problem, featuring Iggy Azalea, which hit number one in the UK in 2014; and Side To Side, featuring Nicki Minaj, which reached number four last year.
She’s currently on a European tour – she’s already played Birmingham and Dublin and is due to be at the O2 Arena in London on Wednesday and Thursday.
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’.”
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.”
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.