Taliban fighters seized a district centre in northern Afghanistan on Saturday, while a bomb at an army base in the southern province of Helmand killed at least five people.
The latest violence came as insurgents pushed ahead with a spring offensive.
Fighters captured the governor's house and police headquarters in Aq Tapa, in the Qala-e Zal district of Kunduz province and Afghan government forces were fighting to drive them out, said Sayed Assadullah Sadat, a provincial council member.
The Taliban's main spokesman Zabihullah Mujahid said fighters had captured a police headquarters, 10 security check posts and a market in Qala-e Zal, north of Kunduz city.
At the other end of the country, in Helmand, insurgents launched a vehicle bomb attack on a military base in Nad Ali district, killing four civilians and a soldier, according to a statement from the provincial governor's office.
The Taliban announced the official start of their annual spring offensive this week, continuing the heavy fighting seen across Afghanistan recently as warmer weather has set in.
Last year, the United States announced a more aggressive strategy with stepped up air strikes and more assistance to Afghan government forces fighting the insurgency, drawing a defiant response from the Taliban, who said they would target American forces in the spring campaign.
The fighting may threaten preparations for parliamentary and district council elections planned in October.
In the eastern province of Nangarhar, a rocket fired by Taliban fighters hit a house in the Hoodi area of Goshta district on Friday night. Five civilians were killed and three were injured.
Attaullah Khogyani, the provincial spokesman for Nangarhar, said three women were among those killed and two women and a child were injured in the attack. Assessments of the size of Taliban territorial control vary but the Pentagon estimates that 56 percent of the country is under government control, while a BBC survey this year estimated the insurgents were active in 70 percent of Afghanistan.