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The Taliban declared the war in Afghanistan was over after their insurgents took control of the presidential palace in Kabul and as US-led forces departed and western nations worked on Monday to move their citizens to safety.
As the militants entered the capital on Sunday, President Ashraf Ghani fled the country, saying he wanted to avoid bloodshed.
Hundreds of Afghans desperate to leave flooded Kabul airport.
"Today is a great day for the Afghan people and the mujahideen," said Mohammad Naeem, spokesman for the Taliban's political office.
"They have witnessed the fruits of their efforts and their sacrifices for 20 years. Thanks to God, the war is over in the country."
It took the Taliban about a week to seize control of the country after a lightning offensive that ended in Kabul as Afghan forces melted away.
The forces had been trained for years and equipped by the US and other western nations at a cost of billions of dollars.
Footage broadcast on TV appeared to show Taliban commanders in the presidential palace with dozens of armed fighters.
Mr Naeem said the form of the new regime in Afghanistan would be made clear soon.
The Taliban do not want to live in isolation and are calling for peaceful international relations, he said.
"We have reached what we were seeking, which is the freedom of our country and the independence of our people," he said.
"We will not allow anyone to use our lands to target anyone, and we do not want to harm others."
A US State Department representative said early on Monday that all embassy personnel, including ambassador Ross Wilson, had been transferred to Kabul airport to await relocation and the American flag had been lowered and removed from the embassy compound.
Hundreds of Afghans gathered on the airport's runways in the dark, pulling luggage and jostling for a place on one of the last commercial flights to leave the country before US forces took over air traffic control on Sunday.
"How can they hold the airport and dictate terms and conditions to Afghans?" asked Rakhshanda Jilali, a human rights activist who was trying to flee to Pakistan.
"This is our airport but we are seeing diplomats being evacuated while we wait in complete uncertainty."
She told Reuters she had received several death threats.
More than 60 foreign countries, including the US, UK, France and Japan, issued a joint statement saying all Afghans and foreign citizens who wanted to leave the country must be allowed to depart.
"The Afghan people deserve to live in safety, security and dignity," they said. "We in the international community stand ready to assist them."
Aid group Emergency said 80 wounded people had been brought to its hospital in Kabul, which was at capacity, and that it was only admitting people with life-threatening injuries.
In a Facebook post, Mr Ghani said he had left the country to avoid clashes with the Taliban that would endanger millions of residents of Kabul.
Some social media users in Kabul said Mr Ghani was a coward for leaving them in chaos. A tweet from the verified account of the Afghan embassy in India said: "We are all banging our heads in shame."
Sharia
In Washington, opponents of US President Joe Biden's decision to end America's longest war, launched after 9/11, said the chaos was caused by a failure of leadership.
Many Afghans fear the Taliban will return to past harsh practices and impose their interpretation of Sharia, or Islamic religious law. During their previous rule between 1996 and 2001, women could not work and punishments such as stoning, whipping and hanging were administered.
The militants have now sought to project a more moderate position, promising to respect women's rights and protect foreign citizens and Afghans.
"We are ready to have a dialogue with all Afghan figures and will guarantee them the necessary protection," Mr Naeem said.
UN Secretary General Antonio Guterres urged the Taliban and all other parties to exercise restraint, and expressed particular concern about the future of women and girls in Afghanistan.
The Pentagon authorised another 1,000 troops to help evacuate US citizens and Afghans who worked for them, a US official said.
A senior US defence official told Reuters on Sunday evening that about 500 people, mostly Americans, had so far been moved to safety, and that the number would increase to 5,000 a day when all planned US forces were in Kabul.
European nations, including France, Germany and the Netherlands, also said they were working to remove citizens, as well as some Afghans who worked fro them.
Russia said it did not think it was necessary to evacuate its embassy for the time being. Turkey said its embassy would continue operations.
American exit
Asked if images of helicopters ferrying personnel were evocative of the departure of US personnel from Vietnam in 1975, US Secretary of State Antony Blinken told ABC News: "Let's take a step back. This is manifestly not Saigon."
Mr Biden has faced increasing domestic criticism after sticking to a plan, initiated by his Republican predecessor Donald Trump, to end the US military mission in Afghanistan by August 31.
On Sunday, Senate Republican leader Mitch McConnell blamed Mr Biden for a "shameful failure of American leadership".
"Terrorists and major competitors like China are watching the embarrassment of a superpower laid low," Mr McConnell said.
Mr Naeem said the Taliban would adopt a policy of non-interference in the affairs of others in return for non-interference in Afghanistan.
"We do not think that foreign forces will repeat their failed experience in Afghanistan once again," he said.
