They are also threatening to destabilise Pakistan, where they control areas in the north-west and are blamed for a wave of suicide bombings and other attacks.
A predominantly Pashtun movement, the Taliban came to prominence in Afghanistan in the autumn of 1994.
It is commonly believed that they first appeared in religious seminaries - mostly paid for by money from Saudi Arabia - which preached a hard line form of Sunni Islam.
The Taliban's promise - in Pashtun areas straddling Pakistan and Afghanistan - was to restore peace and security and enforce their own austere version of Sharia, or Islamic law, once in power.