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.
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.