Taliban militants are trying to seize three major cities from government forces and have been continuing fighting in three major cities of southern and western Afghanistan.
They have entered parts of Herat, Lashkar Gah and Kandahar, reports BBC.
They have made rapid gains in rural areas since it was announced almost all foreign troops would go by September.
But the fate of these key cities could be crucial amid fears of a humanitarian crisis and how long government forces will be able to hold out.
The fundamentalist Islamist militia is already thought to have captured up to half of all Afghanistan's territory, including lucrative border crossings with Iran and Pakistan.
In Lashkar Gah, insurgents were reportedly only a few hundred metres from the governor's office on Saturday - but had been pushed back by nightfall.
It was their second such attempt in as many days. The commander of Afghan forces said they had inflicted significant casualties on the militants on Friday.
The Taliban's firm focus is now on Afghanistan's cities. The situation is fluid but Lashkar Gah, capital of Helmand province where many US and British soldiers lost their lives, looks the most vulnerable right now. Pro-Taliban social media accounts have uploaded videos of their fighters in the heart of the city.
Afghan special forces are being sent in to help push them back, but one local resident told us even if that does end up happening, the Taliban's advance is a powerful assertion of their strength.
The militants are understood to have taken up some positions in the homes of ordinary families, which will make it harder to dislodge them. More lengthy and bloody fighting looks to be ahead.
One MP in Kandahar told the BBC the city was at serious risk of falling to the Taliban, with tens of thousands of people already displaced and a humanitarian disaster looming.
Gul Ahmad Kamin said the situation was getting worse hour by hour, and the fighting within the city was the most severe in 20 years.
He said the Taliban now saw Kandahar as a major focal point, a city they want to make their temporary capital. If it fell, then five or six other provinces in the region would also be lost, Mr Kamin said.
He said the Taliban fighters were on several sides of the city and because of the large civilian population government forces would not be able to use heavy weaponry if the militants got fully inside.
In Herat, a Tolo News reporter said clashes had intensified, with Taliban fighters entering southern parts of the economically important city.
There are reports of fighting in at least five different locations.
The US is still carrying out air strikes to support the Afghan forces, who have recaptured a district around the airport.
A guard outside a UN compound near the airport was killed on Friday in what the UN described as a deliberate Taliban attack.
Residents say few places in the city are safe and some people are taking up arms to defend themselves.
The EU's special envoy for Afghanistan, Tomas Niklasson, said he believed the war was set to get much worse.
He said he feared the Taliban way of thinking now was "something they had in the past - re-establishing... their Islamic emirate".
And the former head of the British Armed Forces, Gen David Richards, warned the international withdrawal could result in the collapse of the Afghan army's morale, leading to Taliban control and possibly a renewed international terrorist threat.
Humanitarian organisations have also warned of a major crisis in coming months as the Taliban continue their offensive - with a lack of food, water and services, and overcrowding in camps for the displaced.
US troops and their Nato and regional allies forced the Taliban from power in November 2001.
The group had been harbouring Osama Bin Laden and other al-Qaeda figures linked to the 11 September 2001 attacks in the US.
But despite a continued international presence, billions of dollars of support and training for the Afghan government forces, the Taliban regrouped and gradually regained strength.
In February 2020, then-US President Donald Trump and allies agreed to formulate a deal with the Taliban on the withdrawal of international combat forces.
This year, President Joe Biden announced the withdrawal would take place by September.