Witnesses say at least three people have died following desperate scenes at Kabul airport.
Video shows some people trying to cling on to planes as they took off.
With scheduled flights suspended, many foreigners and Afghans are now stranded.
The US and other countries have been rushing to evacuate staff and allies.
On Sunday the Taliban declared victory after Afghan President Ashraf Ghani fled abroad and his government collapsed.
The militants’ return to rule brings an end to almost 20 years of a US-led coalition’s presence in the country.
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