The Taliban are fighting with opposition groups and Afghan army remnants in the Panjshir Valley north of Kabul
They have called on the rebels to surrender. Rebels say they want a decentralised state with power shared between different ethnic groups
The top US general says the US may co-operate with the Taliban in the fight against Islamic State militants.
See: https://www.bbc.com/news/live/world-58279900
UK Foreign Secretary Dominic Raab has said the UK government will "not be recognising the Taliban any time in the foreseeable future"
But he added that "we do see the need for direct engagement" and that co-operation on border crossings was an "important first test"
The Taliban have told the BBC that women may have some role in the country's new government but are unlikely to be ministers.