Follow the latest updates on Afghanistan here
Videos on social media showed hundreds of people swarming the tarmac as US military transport aircraft took off, with some even clinging on to the planes as they sped down the runway.
Other videos showed people falling from the aircraft as they climbed away from the Afghan capital.
With land borders now under the control of the insurgents, the airport is the last remaining exit point and there are fears that option may close soon.
US soldiers sent in to try to provide some security at the airport killed two armed men who brandished weapons at them, American media reported.
The US is sending another 1,000 troops to help safeguard the airport, Pentagon spokesman John Kirby said, aimed at bringing a semblance of order to evacuations.
Mr Kirby said the US had suspended flights until the situation could be brought under control.
In a video for local news, former president Hamid Karzai and Abdullah Abdullah appeared together in a video and spoke of efforts to bring the situation in Afghanistan back to normal.
Mr Karzai and Mr Abdullah said that along with Gulbuddin Hekmatyar, they are in talks with the Taliban leadership to achieve countrywide peace and security. Their main effort is to maintain calm in Kabul, they said.
US President Joe Biden was due to give a national address on the crisis in Afghanistan on Monday.
German Chancellor Angela Merkel said her country must urgently evacuate as many people as it could from Afghanistan, for whose safety it is responsible.
Speaking at a closed-door meeting of her Christian Democrat party on Monday, Ms Merkel said: "We are witnessing difficult times. Now we must focus on the rescue mission."
Those needing to leave included 2,500 Afghan support staff as well as human rights activists, lawyers and others whom the government sees as being at risk if they remain in Afghanistan, up to 10,000 altogether.
Foreign Minister Heiko Maas said Germany was seeking to evacuate as many people at risk as it could, and that Nato allies had misjudged the situation when they thought Afghan government forces could hold back the Taliban unaided.
"We want to get as many people out of the country as quickly as possible," Mr Maas told reporters outside the Foreign Ministry.
The Afghan ambassador to the UN, Ghulam Isaczai, called for a humanitarian corridor to be set up to allow Afghans to leave the country at an emergency UN Security Council meeting on Monday.
He said he was "speaking on behalf of millions of people in Afghanistan whose fate hangs in the balance" after the Taliban takeover.
The UN must call on the Taliban to cease targeted killings and revenge attacks, he said.
He asked the UN not to recognise a Taliban government and urged the international community to help humanitarian support reach the Afghan people.
British Defence Secretary Ben Wallace said the Taliban takeover was a "failure of the international community" and that Western military intervention in the country was incomplete.
"All of us know that Afghanistan is not finished. It's an unfinished problem for the world and the world needs to help it," he told the BBC.
HR McMaster, a former US national security adviser who was sacked by Donald Trump in 2018, accused his country of "wilful ignorance" over its failure to realise the Taliban would swiftly take control of Afghanistan.
Mr Wallace and Mr McMaster strongly criticised of a deal agreed to by Mr Trump during his time as president.
Under the deal, the US was to withdraw all of its troops by May 2021 in exchange for security guarantees from the militants.
The deal weakened the Afghan government and security forces and strengthened the Taliban, Mr McMaster said.
Meanwhile, Bahrain announced it would initiate consultations with the other Gulf Arab monarchies regarding the situation in Afghanistan in its capacity as current head of the Gulf Co-operation Council.
"The council of ministers has tasked the foreign minister to co-ordinate and consult with the GCC states regarding the developments in Afghanistan, in the framework of Bahraini presidency" of the group which also includes Saudi Arabia, the United Arab Emirates, Qatar and Oman, the kingdom's government media office said on Twitter.