The blasts, which also wounded 80 people, came as officials tallied votes from April 30 parliamentary polls amid a protracted surge in bloodshed that has killed more than 3,300 people this year.
The government has blamed external factors, such as the civil war raging in neighbouring Syria, for the escalating unrest this year.
But analysts and diplomats say the Shiite-led authorities must also do more to reach out to the disaffected Sunni minority and undermine support for militancy.
Two of them exploded near a traffic police headquarters in the eastern Baladiyat neighbourhood, and blasts also hit Sadr City, Urr, Jamila, Maamal and the central commercial district of Karrada.
Another vehicle rigged with explosives was also detonated in the mostly-Sunni area of Arab Jubour, killing three, while a roadside bomb also went off near a police patrol in west Baghdad, killing one.
Smoke could be seen rising above several areas of the capital, and AFP journalists reported several shopfronts badly damaged and nearby cars reduced to mangled wrecks of metal.
"He said he would leave the car and go looking for spare parts, and then he left," said 54-year-old Abu Nuri.
"Only one of my employees was in the shop when a huge explosion went off. He fell down, and smoke was everywhere. Many people were crying, and others were running away."
Abu Nuri railed against the authorities and security forces, telling AFP: "The state has failed, it has completely failed!"
Elsewhere, a young boy was killed in a rocket attack just north of the capital.
No group immediately claimed responsibility for the apparently coordinated attacks.
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