On Monday, militants stormed a government building in the east, killing two police guards and five civilians.
They also struck a police checkpoint to the south and killed nine policemen.
Also on Monday, rockets hit inside the grounds of the Kabul international airport but caused no damage. Rockets also struck a NATO base at Bagram, just north of the Afghan capital, causing minor damage, the alliance said.
This year's Taliban spring offensive comes at a sensitive time, against the backdrop of a key presidential election. Militants have also stepped up terror attacks to sow insecurity and weaken the government as international forces prepare to withdraw from the country by the end of this year.
Monday's attack on the provincial justice ministry building in the city of Jalalabad began around 9 a.m., just as employees were arriving for work, said Nangarhar provincial government spokesman Ahmad Zia Abdulzai.
According to the spokesman and the provincial police chief, three attackers shot and killed the two police guards, broke into the ministry and took over the building. One of the attackers died when he detonated his explosives' vest inside the ministry, while the other two were killed by police, said Gen. Fazel Ahmad Sherzad, Nangarhar's police chief.
Taliban spokesman Zabihullah Mujahid claimed responsibility for the attack in an email to reporters. He said the attack was retaliation for what he called harsh rulings by the justice ministry against the Taliban. Last week, the Taliban said they would launch their annual spring offensive on May 12.
In southern Helmand province - the Taliban heartland - insurgents killed nine policemen in an attack on a checkpoint in the Sangin district, district governor Sulaiman Shah Sarwani said.
Also on Monday, a rocket hit a market in the Siagred district of Parwan province north of Kabul, killing two civilians and wounding four. The Taliban also claimed responsibility for that attack.
Elsewhere in the east, groups of militants, including some on motorbikes, attacked police checkpoints on the outskirts of the city of Ghazni, killing three people and wounding eight, said the provincial governor, Mohammad Ali Ahmadi. Two women and a policeman were killed, while two policemen and six civilians, including three children, were wounded, Ahmadi added.
No one immediately claimed responsibility for the Ghazni attack.
Elsewhere in Jalalabad, attackers targeted a police vehicle and detonated a roadside bomb, wounding six people, including two policemen.
In the capital, Kabul, two rockets fired from unknown location landed inside the perimeter of the city's international airport without causing any damage, Kabul police spokesman Hashmat Stanikzai said.
To the north of the city, four rockets struck the Bagram military airbase, 48 kilometers (30 miles) from Kabul. The projectiles hit the Bagram Airfield around 3 a.m., resulting in minor damage to equipment and a building, said the international coalition.
RA/SHI