The bomb was inside a truck which blew up in the Jamila market in Baghdad’s Sadr City district.
Police officer Muhsin al-Saedi said: “A refrigerator truck packed with explosives blew up inside Jamila market at around 6 am (03:00 GMT).
“Many people were killed and body parts were thrown on top of nearby buildings.”
The food market in the northeastern suburb is one of the biggest in Baghdad.
Angry people gathered at the site of the explosion, some crying and shouting the names of their missing relatives and others cursing the government.
“We hold the government responsible, fully responsible,” witness Ahmed Ali Ahmed told Reuters.
The attack is one of the largest to hit Baghdad since Haider al-Abadi became prime minister a year ago.
Rescuers pulled bodies from the rubble as the blast burned much of the market, leaving charred wooden market stalls and scattered fruit and vegetables across the ground. At least 200 people were wounded and taken to local hospitals in ambulances or their own cars.
In a statement posted online, Islamic State said it had targeted the army and militia fighters in the Shi’ite neighbourhood.