Neighbouring countries such as Poland, Moldova, Hungary, Romania and Slovakia have all made an unconditional welcome to those feeling the Russian invasion.
More than half a million people have fled Ukraine since Russia began its invasion last week, the head of the UN's refugee agency (UNHCR) said on Monday.
Some walked many miles through the night while others fled by train, car or bus, forming lines kilometres long at border crossings with neighbouring countries: Poland, Moldova, Hungary, Romania and Slovakia. Some even fled into Belarus.
Those arriving were primarily women, children and the elderly after Ukrainian President Volodymyr Zelenskyy prohibited men of military age from 18 to 60 from leaving.
Filippo Grandi, the UN High Commissioner for Refugees, said more than 500,000 had left Ukraine in the four days since Russia's invasion.
On Sunday, the UNHCR said 45,200 refugees had arrived in Poland in just 15 hours.
The agency's spokesman, Chris Melzer, said there were queues of 14 kilometres at the Poland-Ukraine border and waiting times of 40 hours.
The agency expects up to four million Ukrainians could flee if the situation deteriorates further.
'We're letting everyone in'
In contrast to other conflicts around the globe, Russia's attack on its Western neighbour has ignited a massive outpouring of support for the fleeing Ukrainians.
This included an unconditional welcome from nations like Poland and Hungary that did not want to accept those fleeing conflict and poverty in the Middle East and Africa.
Poland declared its border open to fleeing Ukrainians -- even those without official documents -- and dropped its requirement to show a negative COVID-19 test.
On Saturday, the Polish authorities sent a hospital train to pick up those wounded in the war in Mostyska, in western Ukraine, and bring them to the Polish capital of Warsaw for treatment.
The hospital train left the border town of Przemysl with five carriages to transport the wounded and four others stocked with humanitarian aid for Ukraine's Lviv district.
Poland's government said Saturday that more than 100,000 Ukrainians had crossed the Polish-Ukrainian border in the past 48 hours alone.
Even Hungarian Prime Minister Viktor Orbán, Europe's leading anti-migrant politician, travelled to the border town of Beregsurány, where he said Hungary was accepting all citizens and legal residents of Ukraine.
"We're letting everyone in," Orbán said.
Regular people were also opening their homes to refugees and volunteering at welcome centres. In Poland, a Facebook page was set up to offer people rides in private cars from the border and other help.
A warm welcome in Romania
In Siret, Romania, the border post was crowded with Ukrainians on Saturday, and humanitarian groups set up tents a few miles in, offering food and drink to those arriving.
"The feeling is like you are in a train station," Euronews reporter Claudiu Popa said. "Everybody is coming, everybody is leaving, everybody is going somewhere."
But a makeshift welcoming committee made up of Romanian authorities, NGOs, church groups and private citizens was there to meet with the Ukrainian refugees.
"From the very first moment they pass the border, somebody is starting to ask, 'are you thirsty, do you need anything, would you like some free accommodation here in the county, or do you need to get to the capital, to Bucharest? Would you like something to drink, something to eat, would you like some small toys for your children?'
Most people don't want to stay in the area, Euronews was told. The final destinations are Lithuania, Germany, Poland, or Italy for some.
Others in Europe keen to help
Other countries in Europe have already either announced their willingness to help or begun taking in refugees from Ukraine.
In Serbia, Minister of Foreign Affairs Nikola Selaković said on Saturday that the country is ready to accept refugees and that, according to its capacities, send humanitarian and medical aid to Ukraine.
Serbia is one of a handful of European countries that has not introduced sanctions against Russia, but its President Aleksandar Vučić said that it "supports the territorial integrity of Ukraine" and that its willing to assist in any humanitarian efforts.
Albanian Prime Minister Edi Rama also confirmed that Albania is ready to receive Ukrainian refugees fleeing war.
"We will welcome people, family members and those who are leaving or will leave Ukraine," he said.
Croatia has also opened its doors to Ukrainians, taking in about a dozen people on Saturday. The Croatian Red Cross and other authorities are drawing up plans to host incoming refugees in Zagreb and Osijek, local outlets reported.
In Bosnia and Herzegovina, citizens are offering to host refugees at their homes on social media. "We've been through the same… and we're more than happy to help," one family posted on Facebook.