Two roadside bombs exploded in the main Shurja market in central Baghdad within minutes of each other, killing 10 civilians and injuring 50, Iraqi police spokesman Bilal Ali Majid said.
Three more bombs had exploded earlier on Tuesday near the interior ministry building in central Baghdad, killing 9 people and injuring eight.
The injured included four civilians and four policemen.
Both Shurja market and the interior ministry in Al-Nahda building are in central Baghdad, which has been the focus of growing sectarian violence between Shiias and Sunnis that many fear could force the country into civil war.
The blasts came as the United States boosts its troop levels in the Iraqi capital in an effort to stop insurgent and sectarian violence, which has raised fears of full-blown civil war.
The violence is claiming around 100 lives every day and undermining confidence in Iraq's new Shia-led government.
In a separate incident, gunmen in two cars stormed a bank in Azamiyah district of Baghdad and killed five people - three guards and two bank employees.
The gunmen drove away with 7 million Iraqi dinars (less than $5,000).
Elsewhere, the bullet-riddled corpses of seven Iraqi border guards were found in Kharqush, a village on the Iranian border.
In Tikrit, a town north of Baghdad, one policeman was killed and three others wounded when their patrol was struck by a bomb, police said.
The government of Nuri al-Maliki, the Iraqi prime minister, has vowed to confront militias, but some of the armed groups have close ties to parties in the administration.
The government of Nuri al-Maliki has
vowed to confront militias
Donald Rumsfeld, the US defence secretary, has extended the tour of duty for about 3,700 troops from the 172nd Stryker Brigade, based in Mosul, so they could be sent to Baghdad. The troops began arriving in the capital on Sunday.
General George Casey, the top US officer in Iraq, told a news conference on Monday that the US and Iraqi troops would drive guerrillas and militia "death squads" from Baghdad and improve security by Ramadan, which falls in late September this year.
But a security crackdown in the capital, ordered by the government and involving about 50,000 US-trained Iraqi security force personnel, has failed to ease bloodshed.
On Monday, at least 28 people were killed and 64 wounded in attacks around the country.