Bombs were set off in several Baghdad neighborhoods, including Sadr City, and near the heavily fortified Green Zone where several Western diplomatic missions are located, Voice of America reported.
No one immediately claimed responsibility for the attacks, officials said.
The violence came on the eve of the 10th anniversary of the U.S.-led invasion, and a month before Iraq is scheduled to conduct its first elections in three years.
In Anbar and Ninevah provinces, however, the elections scheduled for April 20 will be delayed up to six months because of threats and the killing of several candidates, the prime minister's office said.
An Interior Ministry official said car bombs, sticky bombs and guns were used in the attacks, China's state-run Xinhua news agency said.
Baghdad Operations Command said forces defused five car bombs and a number of explosive vests throughout the capital, IraqiNews.com reported.
In Iskandariyah, south of Baghdad, a suicide bomber drove an explosive-laden truck into the entrance of an Iraqi army base, killing at least two people and wounding more than 10 others, a local police official told Xinhua.
Another car bomb detonated in Iskandariyah, wounding 10 people, the official said.
Officials said at least four soldiers were wounded in a car bomb explosion near their patrol in Tarmiyah, north of Baghdad.