Afghan government forces stay vigilant in Kabul.

Afghan government forces struggled against Taliban assaults on several major cities on Sunday as Taliban fighters struck Kandahar airport in southern Afghanistan with at least three rockets overnight.

The insurgents stepped up a nationwide offensive that saw a key airport in the south come under rocket fire overnight that the aim was to thwart air strikes conducted by Afghan government forces.

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"Kandahar airport was targeted by us because the enemy were using it as a centre to conduct air strikes against us," Zabiullah Mujahid, a Taliban spokesperson, said to the media.

Taliban has now started to besiege provincial capitals.

Fighting has surged in the months since early May when US-led foreign forces began a final withdrawal from Afghanistan that is now almost complete.

After seizing large tracts of rural territory and capturing key border crossings, the Taliban has now started to besiege provincial capitals.

Flights out of Kandahar, Afghanistan's second-largest city, were halted after rockets struck the airport before dawn.

Afghan government officials said the rocket attacks forced authorities to suspend all flights and that the runway was partially damaged. There were no immediate reports of casualties, the officials said.