No group has claimed responsibility for the attacks, but residents of the city blame the Islamist sect Boko Haram, based in nearby Maiduguri, where a suicide blast earlier on Friday damaged a military headquarters.
Militants from Boko Haram have targeted police and military, community and religious leaders, as well as politicians, in scores of attacks in recent months.
In a mainly Christian neighbourhood of Damaturu called Jerusalem, six churches were bombed in addition to a police station.
Soldiers and police have mounted checkpoints in parts of the city, searching vehicles and carrying out pat-downs of drivers and passengers.
In the outlying town of Potiskum, a grenade narrowly missed a police station and an ensuing gun battle left at least one policeman dead.
The string of attacks came two days ahead of the annual Muslim celebration of Eid al-Adha, or the Feast of Sacrifice, and police have been put on red alert nationwide.
Nigeria's north is predominantly Muslim, with pockets of Christian communities.
The sect, which wants to see the strict application of Sharia or Islamic law in northern Nigeria, staged an uprising which was brutally put down by security forces in 2009.