The attacks come as security forces and allied Shiite militiamen and Sunni tribesmen fight to regain ground from militants led by the Islamic State (IS) jihadist group who overran swathes of the country in June.
In Baghdad, a car bomb exploded in the Shiite-majority Karrada district, killing at least eight people and wounding more than 20, while a roadside bomb in the Bayaa area killed four and wounded at least 11.
South of the capital, another car bomb near a market in Mahmudiyah killed at least two people and wounded seven, while a motorcycle bomb in central Kirkuk in the country's north killed eight and wounded 13.
Accounts of those attacks varied and many of the victims were civilians, but two security officials said an army intelligence base where senior militants are jailed was among the targets.
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In addition to the blasts and shelling, two more suicide bombers were arrested near the base before they could attack, officials said.
Sunni extremists often target members of Iraq's Shiite majority, whom they consider heretics, and also frequently attack security forces.
The United States has carried out a more than five-week air campaign against the IS jihadists, while France has announced it will take part in strikes on them in Iraq as well.