The militants are on the brink of taking total control now that US-led forces have all but withdrawn.
Taliban fighters have entered Afghanistan’s capital Kabul, after President Ashraf Ghani fled the country.
The militants are on the brink of taking total control, after rapidly seizing territory as US-led forces withdrew after 20 years on the ground.
Reports say the Taliban have seized the presidential palace.
Thousands of Afghans have sought refuge in Kabul in recent weeks and there were scenes of panic in the city on Sunday.
A Taliban spokesman told the BBC there would be “no revenge” on Afghans.
Western countries have been scrambling to evacuate their citizens. The US sent military helicopters to transport staff from its heavily fortified embassy compound to the airport.
However, the US Embassy later said there were reports of gunfire at Kabul’s airport. It warned US citizens in the area to take shelter as “the security situation… is changing quickly”.
It is almost 20 years since the Taliban were removed from power by a US-led military coalition.
US President Joe Biden has defended the withdrawal of American troops, saying he could not justify an “endless American presence in the middle of another country’s civil conflict”.
Taliban spokesman Zabihullah Mujahid said militants had been ordered to enter parts of Kabul on Sunday, after previously holding back at the outskirts of the city.
He said Taliban forces were going in to prevent chaos and looting after security forces left parts of the city and their checkpoints.
Taliban fighters were ordered not to harm civilians or enter homes, he said, adding: “Our forces are entering Kabul with all caution.”
A senior interior ministry official told Reuters news agency that Taliban fighters had reached Kabul “from all sides”.
The Taliban advance into Kabul came as officials told reporters that President Ghani had fled. Details of his whereabouts remain unknown, but some reports said he was heading for Tajikistan.
“God will hold him accountable and the nation will also judge,” top Afghan official Abdullah Abdullah said.
There were reports of fighting and casualties in the city’s Qarabagh district.
Thousands of Afghans have sought refuge in Kabul in recent weeks after fleeing violence elsewhere in the country.
There was panic in the city as the Taliban drew closer to victory.
Some residents have been trying to reach the airport to leave the country. Cars have been abandoned and people have opted to walk because of traffic jams.
One 22-year-old student told the BBC that he had walked more than five hours to reach the airport.
“My feet hurt, they have blisters and I’m finding it difficult to stand,” he said.
“It was like a military town – people were in traditional clothes, but they had weapons and were firing in the air. It reminded me of the Jihad that I heard of from my parents.”
People were also rushing to withdraw cash from ATMs, and queuing to get travel documents at the passport office and at foreign visa centres.
Farzana Kocha, an MP in Kabul, told the BBC that people did not know what to do, with some running or hiding in houses.
Taliban spokesman Suhail Shaheen told the BBC that people in Kabul had no need to worry and that their properties and lives were safe.
“We are the servants of the people and of this country,” he said.
He added that the group did not want Afghans to flee, but instead to stay and help with the post-conflict reconstruction.
The US has deployed 5,000 troops to help remove its staff and the Afghans who assisted with its mission. Helicopters transporting embassy personnel could be heard over the city, and there were reports of smoke rising near the embassy compound as important documents were destroyed.
About 600 British troops are being deployed to assist with their own withdrawal mission.
UK Foreign Secretary Dominic Raab said he had “deep concerns” about the future of Afghanistan.
“It is critical that the international community is united in telling the Taliban that the violence must end and human rights must be protected,” he tweeted.
Other countries are also evacuating their nationals, scaling back their presence in Afghanistan and in some cases closing their embassies altogether.
Canada has temporarily closed its embassy, and a Nato official said several European Union staff had been moved to an undisclosed location in Kabul.
Russia is planning to convene an emergency meeting of the UN Security Council to discuss the situation in Afghanistan.
It says it will not be closing its embassy, because it has been provided with security assurances by the Taliban.
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