A total of five policemen were killed in the attacks.
Police said the first explosion occurred in the city's Tisaeen market, an area largely inhabited by Shi'ite members of the city's Turkmen community, killing 13 people, including two policemen, and wounding 18.
About half an hour later, a suicide car bomb attempted to ram the main police headquarters, prompting police to open fire on it.
The car exploded, missing its target, said police commander Major General Torhan Yussef, but two policemen died and eight civilians were wounded.
The three other car bombs, one of which was a suicide attack, exploded in different parts of the city, killing three more policemen and wounding more than a dozen people, including Colonel Taher Salah al-Din, the police chief for the southern Kirkuk Hurriyah neighbourhood.
The bombing campaign comes in the wake of the arrest of eight suspected insurgents by US and Iraqi forces in two operations on Sunday.
In the aftermath of Wednesday’s killing of insurgent leader Abu Musab al-Zarqawi, his followers have vowed campaigns of revenge.
Yesterday saw a number of bomb attacks in Mosul, Tall Afar and Baghdad.
A centre of oil production and home to a complex ethnic and sectarian mosaic of Kurds, Turkmen and Arabs, Kirkuk has been largely spared the large car bombs which has beset cities like Mosul and Baghdad.