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The UN has said more than 500,000 refugees have fled Ukraine for neighbouring countries after Russia's incursion, with the Polish government saying 280,000 people had arrived there.
They have endured days-long journeys into Poland in often freezing temperatures, with most entering the EU with little more than backpacks, if that.
The migrants are almost entirely Ukrainian women and children, and those who were in Ukraine but are not from the country.
Ukrainian men aged between 18 and 60 have been blocked from the leaving the country as the government launches a conscription drive to bolster its forces against Russia.
While each migrant is fleeing the conflict, all have their own stories.
Mohamed Cherdoud
Mr Cherdoud, 31, a chef from Algeria, studied in Ukraine but eventually married a Ukrainian woman and lived in Kharkiv, a city less than 50 kilometres from Russia.
He said that over the weekend, an Algerian friend, Abdul Moneim, was killed in Kharkiv after he was hit by a Russian sniper.
Kharkiv, Ukraine's second largest city, has been devastated by attacks after Russia's incursion on February 24.
Mr Cherdoud, who works at a Japanese and Italian restaurant, left two days later and initially planned to build up supplies and hunker down in Kharkiv.
“I wanted to stay there but it was getting worse and worse,” he said at the train station in Przemysl, Poland, less than 10km from the Ukrainian border.
His plan to flee Kharkiv by car ran into problems when his vehicle was damaged by a Russian strike.
“I decided to leave, but the car was hit by an attack so we went to the train station," Mr Cherdoud said. "My home is near an army training school. The Russians were shooting at it and my car got hit.”
Recalling the initial day of the Russian assault, he said: “I was very scared. They were shooting and there was a lot of noise in the sky. Everybody was looking at the sky and checking where the bombs were falling.”
Mr Cherdoud will now go to Warsaw. His wife was on holiday in Egypt when the attack happened. But he wants to return to Kharkiv as soon as he can.
“I have there my friends, my work. I want to live there, I love Kharkiv and in the future, if the war ends, I will go back there.”
He left Kharkiv by train and ended up in Lviv, a city in western Ukraine not far from the Polish border.
“Everybody has a different story,” Mr Cherdoud said.
He and a group of friends took a taxi from Lviv, aiming for the Polish border, but then hit the long queues that have built up as hundreds of thousands try to leave Ukraine.
Mr Cherdoud then walked 20km after the driver said he would go no further, boarding a train at the final station in Ukraine.
“We got on a train for 14 hours," he said. "It was overcrowded. There were little children, everybody was standing.
"Around midnight the train left the station and in eight minutes, it arrived in Przemysl."
Oksana Potelchak
A translator from Kiev, Ms Potelchak left the capital with her two children and sister last Monday, three days before Russian President Vladimir Putin officially announced the "military operation" in Ukraine.
“My husband, he is military. He is in Kiev now," she said. "Every day they have very difficult situations, attacks — but they stand and they are sure that we will win.”
Ms Potelchak said her husband told her Ukrainian forces would not just beat back the Russian offensive on Kiev, but would win across the country.
She did not expect Russia to attack the capital. “We thought maybe the south, the east, but not Kiev. We didn’t expect that.
“We were just lucky" to leave Kiev before the invasion, Ms Potelchak said.
“My sons go to school and everything that we’ve heard from classmates, from their parents — it was horrible.”
While initially some had been able to stay in Kiev, “now many people are still there but they have no chance to leave the city because of the danger".
Ms Potelchak said she has been able to speak to her husband every day.
“My husband told me that I won’t recognise Kiev," she said. "I can’t imagine what’s there.
"I think that our army is very strong and they will do everything so that we get all our cities and our countries back.”
Ms Potelchak describes her overriding emotions as “horrible.”
“Because when we left Kiev, my grandmother stayed there," she said. "She didn’t want to leave.
"But the parents of my husband have taken her to their town near the Kiev region. It is very difficult to describe the emotions. I think it’s even impossible. I wish that no one in the world has such an experience.”
Now she will go to Germany, to support other arriving Ukrainians.
“I’m an interpreter and I speak German," Ms Potelchak said. "I think it will be easier for the first time when people come [from Ukraine].
"The volunteers in Germany, they try to develop different initiatives for people who come from Ukraine.
“My sister is also in Germany. We are trying to help Ukrainians who expect to spent maybe one, two, three months in this not very pleasant situation. But we have to do it for our children.”
Tatiana and Fatih Yarkup
Tatiana and her partner Fatih Yarkup, 26, lived in Odessa, a port city on the Black Sea.
After the Russian assault, they woke to the reports of the conflict. They left last Friday.
“The bombs were falling and we thought the shooting and the war would come to Odessa soon, so we decided to leave before it does,” Tatiana said.
“The days were OK but the nights were terrible. Odessa is a port city and they were shooting from the sea.
"There was nowhere to go, it was scary to stay at home. There was nowhere to hide, not like in Kiev because Odessa doesn’t have subway system.”
They managed to drive most of the way, about 7km from the border, but the car broke down.
“We were waiting in line and had to start the car and stop the engine over and over again, so the battery stopped working and we had to walk,” Mr Yarkup said.
The couple said it had been hard to get fuel because of attacks on petrol stations.
“We slept in our car," Mr Yarkup said. "It was very scary because we were sleeping in the fields and the roads were being bombed and also the petrol stations.
"We managed to almost get to the border crossing but we stopped 7km from the crossing.”
Mr Yarkup is in the third year of his medicine degree.
“On Wednesday I was studying at the university and on Thursday the war started,” he said.
They have been told they can get a train ticket to anywhere in Poland, but only have a limited network to rely on.
“We just arrived here. We don’t know what we are doing or what we should do,” Tatiana said.
Lyudmila
Lyudmila, her sister, son and mother left Kiev four days ago by car. Her husband, who stayed in Ukraine to fight, drove them as far as he could but the traffic jams saw them at a virtual standstill.
“We’d go five metres and then have to wait half an hour,” Lyudmila said.
They relied on lifts for the rest in way, because her 70-year-old mother could not walk far.
Now the family are going to Warsaw, but Lyudmila's thoughts are never far from home.
“We pray that the fighting will soon stop, because the people are dying,” she said.
Head of UN: 'The fighting in Ukraine must stop'
