The four separate bombings were a further challenge to the beleaguered government of Prime Minister Haider al-Abadi, who is struggling to prove that his forces can maintain security in Baghdad and elsewhere.
One of the bombings Tuesday struck an outdoor market in the heavily Shiite district of Sadr City, a frequent bombing target.
Fighters with al-Sadr's militia, Saraya Salam, deployed to the streets of Sadr city following the bombing, another indication of a lack of confidence in Iraq's official security forces.
In an online statement, the Islamic State group claimed responsibility only for the deadliest bombing of the day, which took place in Baghdad's northeastern Shaab neighborhood and where at least 34 people were killed and 75 others were wounded.
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The IS statement said the attack was carried out by an Iraqi who targeted members of Shiite militias. The Associated Press could not verify the authenticity of the statement but it was posted on a militant website commonly used by extremists.
Later in the afternoon today, a suicide car bombing hit a crowded outdoor market in Sadr City, killing 18 people and wounding 35 others. Sadr City has been one of the worst-hit districts as the Islamic State group has increasingly employed terrorist attacks against Iraqi civilians far from frontline fighting.
Al-Sadr led anti-government protests last month that peaked with the breaching of the Green Zone, the highly fortified compound in the heart of Baghdad that's home to many ministries and foreign embassies.