Police said that nine corpses were pulled out of the Tigris river and turned in to the mortuary in Kut, 160km southeast of Baghdad. They had been blindfolded, had their hands and legs tied, and showed signs of torture.
They also said the body of a civilian kidnapped the night before was taken to the mortuary after being found in an industrial area of the city.
Later, five beheaded corpses which were found on the banks of the Tigris in al-Suwayra, 40km south of Baghdad, were brought to the mortuary.
Police said that five people were killed and six others injured when a parked car bomb blew up near a fuel station in a southwest district of Baghdad.
Ten people were killed and 11 wounded in a shootout near the al-Mishahda Sunni mosque in the northern Huriya district of Baghdad after evening prayers.
Earlier, Captain Jamil Abbas of Baghdad's Major Crimes Bureau was killed when a bomb hidden in his car detonated at 8.30am.
In the capital's Dora district, an official of the neighbourhood power station and a friend were killed, police said. The two were shot by unknown assailants while driving through the area.
In the neighbourhood of Karrada in Baghdad, one civilian was killed and three traffic policemen were injured when a roadside bomb targeting a patrol exploded. Shortly after, another civilian was killed and two more wounded when a second bomb exploded nearby.
In Karma, 80km west of Baghdad, an Iraqi soldier on foot patrol was shot and killed by a sniper.
In clashes between gunmen and an Iraqi army patrol in the northern city of Mosul, one fighter was killed and three others arrested, police said.
Two Iraqi soldiers were also killed and three wounded when unidentified people opened fire on them in their car in an area south of Kirkuk on Tuesday, officials said.