The gunbattles occurred hours after anti-U.S. cleric Muqtada al-Sadr warned he would declare war if the government does not stop a U.S.-Iraqi crackdown against his followers.
Meanwhile, Secretary of State Condoleezza Rice arrived in Baghdad on a visit to promote Sunni and Kurd support for the U.S.-backed Shiite government.
Iraqi police and hospital officials said six people - four men and two boys ages 8 and 10 - were killed in fighting in Baghdad's main Shiite district of Sadr City after midnight.
The U.S. military had no immediate comment on Sunday's fighting but said its forces in Sadr City killed seven armed "criminals" the day before in Sadr City - two in gunbattles and five in two separate airstrikes. The military said it does not engage if civilians are seen in the area.
Clashes also broke out between Iraqi forces and militiamen in the southeastern Shiite district of New Baghdad, wounding three people, including a woman, police said.
A roadside bomb targeting a U.S. military convoy in the eastern Baladiyat area killed a civilian bystander, police said.
Recent fighting between government forces and Shiite militias has largely been centered in Sadr City, a sprawling district of 2.5 million people, and the southern city of Basra.
Iraqi security forces also stormed into the local Sadrist office Sunday morning in Aziziyah, 35 miles southeast of Baghdad, detaining four people and confiscating computer disk drives and documents, police said.
In the warning, posted Saturday on his Web site, al-Sadr said he had tried to defuse tensions by declaring a unilateral truce last August, only to see the government respond by closing his offices and "resorting to assassinations."
He accused the government of selling out to the Americans and branding his followers as criminals.
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"So I am giving my final warning ... to the Iraqi government ... to take the path of peace and abandon violence against its people," al-Sadr said. "If the government does not refrain ... we will declare an open war until liberation."
A full-blown uprising by al-Sadr, who led two rebellions against U.S.-led forces in 2004, could lead to a dramatic increase in violence in Iraq at a time when the Sunni extremist group al-Qaida in Iraq appears poised for new attacks after suffering severe blows last year.
The terror network on Saturday also announced a one-month offensive against U.S. troops and Sunnis who have joined forces with the Americans in a new Internet audiotape by the purported leader of al-Qaida in Iraq, Abu Ayyub al-Masri.
A week of violence has raised concerns that suspected Sunni insurgents are regrouping in the north. U.S. and Iraqi troops have stepped up security operations in Mosul, believed to be one of the last urban strongholds of al-Qaida in Iraq.
Underscoring the danger, gunmen on Sunday ambushed three minibuses carrying university students near Baqouba, 35 miles northeast of Baghdad, and kidnapped their drivers, police said.
The gunmen stopped the buses at a fake checkpoint on a highway but released the students later, a police officer said, speaking on condition of anonymity because he was not authorized to release information.
A rocket slammed into the fortified Green Zone late Saturday, damaging the roof and shattering the windows at a two-story school for Iraqi children who live in the Green Zone, witnesses said.
The school, which has 750 pupils, was empty and no casualties were reported, the headmistress Fawziya Mohammed said.
Internal Shiite tensions have been rising since U.S.-backed Prime Minister Nouri al-Maliki launched a crackdown against militias to break their power and improve security in southern Iraq before provincial elections this fall.
Al-Sadr's followers believe the campaign is aimed at weakening their movement to prevent it from winning provincial council seats at the expense of Shiite parties that work with the United States in the national government.
© 2008 Associated Press