Police say they are trying to establish whether the dead attacker was working alone or as part of a network.
Sixty ambulances attended the incident, and the wounded were taken to eight hospitals around Greater Manchester.
A large cordon remains in place around Manchester Arena and Victoria Train Station, which backs on to the area.
All trains and trams to the station have been cancelled.
The Manchester Arena is the largest indoor arena in Europe, and has a capacity of 21,000.
British Home Secretary Amber Rudd has said the explosion was a "barbaric attack deliberately targeting some of the most vulnerable in our society."
Prime Minister Theresa May has condemned the "appalling incident." She was due to chair a meeting of the Government's emergency COBRA committee on Tuesday morning.