A roadside bomb exploded near an Iraqi army patrol on Saturday in the southern Baghdad neighbourhood of Dora, killing three soldiers and wounding three others, the army said.
At least one civilian was killed and two others wounded in fierce fighting between insurgents and the army in Fallujah, police Lt Mohammed Khalaf said.
Several cars were set on fire in the clash, he said.
Fallujah, 65 kilometers west of Baghdad, used to be the country's main insurgent stronghold until US forces overran the city in November 2004 in the most intense urban combat of the Iraq war.
Now US and Iraqi forces maintain tight control. In the southern city of Basra, four gunmen killed the director of traffic police as he was driving to work, police said. Basra is 550 kilometers southeast of Baghdad.