The attacks, which happened at the same time, included a bomber on a motorcycle, rockets and a roadside bomb which hit a market in the southern Baghdad district of Zaafaraniya, police said earlier on Sunday.
Separately, at least two mortar bombs hit an apartment building in the same area while a separate car bomb exploded.
At least 20 people were killed and 70 wounded after a rocket attack demolished a building and a car bomb hit rescuers rushing to the scene in Baghdad.
The twin explosions knocked down a four-storey building in the Zafaraniyah district of the Iraqi capital, an interior ministry official said.
The official said: "A Katyusha rocket landed on a building in al-Qubaysi market. Five minutes later, 100m away from this building a car bomb went off."
A medical official who saw the attacks said: "There are dozens of bodies in the street. The building just collapsed. It was four-storeys, with homes and shops.
"Civil defence personnel are trying to get bodies out of the building. The shops underneath are destroyed."
A medic at the Ibn Nafis hospital said that a young girl was among the dead.
Three police officers were wounded in a third blast, apparently a roadside bomb, that targeted a patrol heading for the scene, the interior ministry official added.
Meanwhile, a Shia leader has called for neighbourhood committees to provide security in their own districts, casting further doubt on the ability of Iraqi and US forces to reduce violence levels in Baghdad.
Hadi al-Amiri, a member of parliament and head of a Shia militia, said such committees were essential because Iraqi forces still lacked training and were not ready to tackle anti-government fighters.
Armed Shia militias have added
to lawlessness and insecurity
More than 50,000 US and Iraqi forces are taking part in Operation Together Forward aimed at reducing violence levels.
Similar campaigns have failed in the past but Washington hopes to see significant results by the end of September.
"Our forces are not complete to take on this wide terrorism," he said on state television on Sunday.
It is not clear whether the process of forming the committees has already started, although young men in mosques are known to have been approached and asked to join.
In Sunday's other developments, Iraqi and US soldiers hunting for kidnappers raided the health ministry in Baghdad, arresting five suspects.
An official from the US-led force said: "Acting on an Iraqi citizen's tip regarding kidnap victims, Iraqi forces with US advisers searched the ministry of health ... and detained five for further questioning."
The government also announced that it had arrested a 16-member gang alleged to have been plotting attacks on the family of Nuri al-Maliki, the Iraqi prime minister.
"Our forces are not complete to take on this wide terrorism"
Shia militia leader
Following the arrests, however, health ministry workers mounted a small, non-violent anti-American protest outside their offices to demand the release of the detainees, described by one official as guards attached to the ministry.
Qasim Yahiya, a ministry spokesman, denounced the raid and said staff had launched an open-ended strike in protest. He demanded that al-Maliki's cabinet rein in US forces and threatened to lodge a complaint before an Iraqi court.