The deadliest attack took place when twin car bombs hit a police checkpoint near a pet market, killing at least 19 people, officials said.
Police said the attack began when a suicide bomber rammed his explosive-laden car into a checkpoint manned by traffic police in the southeastern neighborhood of New Baghdad, killing five policemen and wounding nine.
Seconds later, a car bomb explosion near the market killed 14 people and wounded 35, police said. Several cars were damaged in the blasts.
In the night, police said a suicide car bomb hit a police post in Ghadeer district in southeastern Baghdad, killing six police officers and two civilians. A sticky bomb attached to a mini-bus also exploded in southeastern Baghdad, killing three passengers, police said.
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Medical officials confirmed the casualties from all attacks. All officials spoke on condition of anonymity because they were not authorised to speak to journalists.
The attacks come after the Islamic State group captured large swaths of territory in western and northern Iraq in a lightning offensive earlier this year, plunging the country into its worst crisis since US troops left at the end of 2011.
The country's once-dominant Sunni minority long has complained of being marginalized and discriminated against, and Sunni grievances with Baghdad are seen as one of the main factors fueling the rise of the Islamic State group.