The interior ministry told the BBC 14 blasts hit various locations, including al-Amil in the south and Halawi and Karrada closer to the centre.
The bombings are the worst in months – and follow the withdrawal of US troops.
They come amid fears of rising sectarian tensions as the unity government faces internal divisions.
It was not immediately clear who was behind the attacks.
The bombs exploded as many people were travelling to work during the morning rush-hour.
Four car-bombs and 10 improvised explosive devices (IEDs) were detonated, officials told the BBC.
A security spokesman in Baghdad, Maj-Gen Qassim Atta, said the attackers had not aimed at security targets.
“They targeted children’s schools, day workers and the anti-corruption agency,” he told the AFP news agency.
Raghad Khalid, a teacher at a kindergarten in Karrada, said “the children were scared and crying”.
“Some parts of the car bomb are inside our building.”
Smoke was seen rising over Karrada district, with ambulances rushing to the scene.
Another woman said her baby had been covered in glass.
“She is now scared in the next room. All countries are stable. Why don’t we have security and stability?” said Um Hanin.
Iraq’s year-old power-sharing government is in turmoil after an arrest warrant was issued for Sunni Vice-President Tariq al-Hashemi on terror charges.
The entire al-Iraqiyya group, the main Sunni bloc in parliament, is boycotting the assembly in protest. It accuses Iraqi Prime Minister Nouri Maliki, a Shia, of monopolizing power.
Tariq al-Hashemi denies the charges. He is currently in Irbil in Iraqi Kurdistan, under the protection of the regional government, but PM Nouri Maliki has demanded that they give him up.
The last American troops departed from Iraq on Sunday, nearly nine years after the war that toppled Saddam Hussein in 2003.
President Barack Obama acknowledged that the situation was not perfect, but said the US forces were leaving behind “a sovereign, stable and self-reliant Iraq, with a representative government elected by its people”.
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