Officials in northern Kunduz province and in Helmand described fierce, well-planned operations, involving a large number of gunmen who attacked under cover of darkness.
Elsewhere in Afghanistan, attacks on civilians and soldiers claimed at least seven more lives today.
The attacks came as President Ashraf Ghani prepared to head to Brussels for a key international aid conference this week, where he expects donors to pledge USD 3 billion a year in assistance for his impoverished, war-torn nation.
Residents and officials said the fighters attacked from all directions today's assault. Mahmood Danish, spokesman for the Kunduz provincial governor, said security forces managed to keep them at bay.
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The Interior Ministry said a policeman was killed and four were wounded in the ongoing fighting. A ministry statement said the situation was being monitored in case reinforcements are needed.
Kunduz is the capital of the strategically important Kunduz province, a breadbasket region that borders Tajikistan to the north and sits on a major crossroad in the country.
Kunduz came under threat again in April, when Afghan forces aided by US troops and air power pushed the Taliban back into the surrounding districts.
In today's attack, the Taliban used residential areas in Kunduz and Afghan "security forces are being very careful to avoid civilian casualties while shooting back at the enemy," said Danish.
The Afghan air force was also supporting the ground forces in the fight, he added.
The US military spokesman in Afghanistan, Brig. Gen. Charles Cleveland said the Kunduz situation was being monitored but that the international alliance was not seeing evidence "to support the reports that Kunduz is under significant attack."
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