According to a broadcast on Al Jazeera, a Taliban spokesman has declared the end of the war, adding that the type of rule the Taliban will pursue, and the set up of the new regime, will soon be made clear.
"We assure everyone that we will provide safety for citizens and diplomatic missions. We are ready to have a dialogue with all Afghan figures and will guarantee them the necessary protection," Taliban political office spokesman Mohammad Naeem told Al Jazeera.
Suhail Shaheen, another spokesman for the militant group, told the AP news agency that the Taliban would hold talks in the coming days toward forming an "open, inclusive Islamic government in Afghanistan."
In live footage aired earlier by Al Jazeera, Taliban commanders and fighters could be seen sitting inside the presidential palace in Kabul and declaring victory in their campaign against Afghan forces.
Taliban members were shown entering the presidential office and taking down Afghan flags.
First evacuations of German citizens taking place
The first evacuation operations of German citizens from Kabul have started early Monday, German news agency DPA reported. Three dozen Germans, along with several Swiss nationals, are being flown to the Qatari capital of Doha.
The evacuees are being transported on US aircraft. The German military has said it is sending more aircraft throughout Monday to carry out evacuations.
All US Embassy personnel safely evacuated: State Department
All personnel at the US Embassy in Kabul have been safely evacuated as of early Monday, State Department spokesperson Ned Price said.
"All Embassy personnel are located on the premises of Hamid Karzai International Airport, whose perimeter is secured by the US Military," Price said in a statement.
Germany, other countries, call for safe evacuations
Over 60 countries, including Germany and the EU, urged all parties in Afghanistan in a joint statement released early Monday to allow the "safe and orderly departure" of foreign nationals and Afghans from the country amid a rapidly deteriorating security situation.
The countries said "roads, airports and border crossing must remain open, and calm must be maintained."
"The Afghan people deserve to live in safety, security and dignity," the statement added. "We in the international community stand ready to assist them," the statement said.
A US official told Reuters news agency on Monday that most Western diplomats have left Kabul.
US taking steps to secure evacuations from Kabul airport
A joint statement from the US Department of State and Department of Defense said the US will take over air traffic control at the international airport in Kabul, as Americans and Afghans are evacuated from the city.
Images on social media showed chaotic scenes as people tried to board aircraft out of the country.
Thousands of US citizens will be flown out of the country in the coming days, along with Afghan nationals who worked for the US embassy in Kabul and their families.
The statement also said the US is accelerating the evacuation of Afghans who are eligible for Special Immigrant Visas.
US authorizes another 1,000 troops to aid evacuations
The US is sending another 1,000 troops to support evacuations from Kabul airport amid reports of scattered gunfire, officials said.
According to a US defense official cited by the AP news agency, the soldiers were heading directly to Kabul instead of going to Kuwait as a standby force.
Saudi Arabia evacuates embassy
Saudi Arabia has evacuated all members of its diplomatic mission in Kabul, the Saudi Foreign Ministry said.
"All members of the Kingdom's embassy in the Afghan capital, Kabul, have been evacuated, and they have arrived home," the ministry said in a statement.
British troops arrive in Kabul
Britain's Defense Ministry said UK troops have landed in Kabul to help the evacuation mission.
Johnson said British nationals were the priority, as well as Afghans who helped British forces in Afghanistan over the past 20 years.
"The ambassador is working round the clock, has been there in the airport to help process the applications," Johnson said after chairing an emergency Cabinet meeting.
The British Parliament will be recalled from the summer break for one day on Wednesday to address the government's response to the crisis.