Survivors say the fire started after Goodbye to Gravity heavy metal band set off fireworks, causing the ceiling and a pillar to catch alight and producing heavy smoke.
A stampede for the exit followed. At least 184 people are being treated in Bucharest hospitals, some with severe burns.
Romanian government has declared three days of national mourning.
“I am saddened, but also revolted that a tragedy of this scale could have taken place in Bucharest,” President Klaus Iohannis said.
“We already have indications that the legal regulations had not been respected,” he said.
“I hope that the authorities manage their inquiry with speed and rigor.”
The fire broke out at about 22:30 at the Colectiv club, which was hosting a free rock concert by the band Goodbye to Gravity. Up to 400 people are thought to have been inside.
Some people initially thought the fire was part of the show, survivors said.
The Colectiv club was in a converted former factory with two small exit doors, only one of which was possible to open initially.
More than 180 people suffered injuries, including from burns and smoke inhalation. Officials have warned that the death toll may rise.
Hundreds of people who responded to appeals for blood donations queued outside hospitals and blood banks on October 31.
PM Victor Ponta cut short a visit to Mexico to return to Bucharest.
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