Taliban militants have attacked a Nato military outpost near the airport in the eastern Afghan city of Jalalabad.
The Nato-led International Security Assistance Force (Isaf) said the attack on its forward operating base in Nangahar province lasted two hours.
Eight militants, one of whom was wearing a suicide belt, died. The Taliban said it was behind the assault.
Later, two policemen and six civilians were killed by a bomb on a motorbike in Kunduz province, local officials said.
Another 18 people were wounded in the attack on a crowded market, which appeared to target a local militia leader in Emam Sehab district.
In a separate incident in southern Afghanistan, Isaf reported that three Nato soldiers were killed in an insurgent attack.
No further details were given about the incident, including the nationalities of those killed.
'Showered with bullets'
The attack near Jalalabad's airport began at about 0530 (0100 GMT) on Saturday, the second time in six months that the area has been targeted.
Witnesses said they heard explosions and saw smoke rising in the area.
An Isaf statement said that Forward Operating Base Behsud "received small arms fire from an unknown number of insurgents".
It said the assault was repelled by Isaf and Afghan troops, who sustained no casualties. Helicopters were also called in.
Afterwards, a nearby residential district was showered with bullets, witnesses said, and a number of bodies were seen lying on the ground.
Taliban spokesman Zabihullah Mujahid said 14 suicide bombers had been involved.
"They entered the airport. Some of them have blown themselves up," he was quoted as saying by AFP news agency.
Isaf said four of the attackers were dressed in Afghan army uniforms.
The attacks come a day after a suicide car bomber struck a convoy of Isaf troops in Kabul. The explosives were detonated before the car reached the convoy, killing one Afghan civilian.