More than one million people have fled Ukraine since Russia invaded last week, according to the United Nation's refugee agency.
UNHCR spokesman Chris Melzer described the figure as "unprecedented in Europe since World War II".
UN High Commissioner for Refugees Filippo Grandi has called for a ceasefire in Ukraine "so that life-saving humanitarian assistance can be provided".
UNHCR has already warned that if Russia continues to prosecute its war the ultimate number of refugees could exceed 4 million people - 10% of Ukraine's population.
On Tuesday the UN launched an emergency appeal for $1.7 billion to provide urgent humanitarian aid to people caught up in the Russian invasion and for the refugees fleeing the violence.
"We are looking at what could become Europe's largest refugee crisis this century," Mr Grandi said.
In just seven days we have witnessed the exodus of one million refugees from Ukraine to neighbouring countries.
— Filippo Grandi (@FilippoGrandi) March 2, 2022
For many millions more, inside Ukraine, it's time for guns to fall silent, so that life-saving humanitarian assistance can be provided.
The first wave of people fleeing across Ukraine's borders were likely to be people with cars, resources and some connections in other European countries, he said.
But as Russia's offensive continues, people who are "more vulnerable" could start to flee, he told reporters.
The UNHCR also noted that an additional 96,000 people had crossed into Russia from the separatist eastern Donetsk and Luhansk regions between 18 and 23 February, before the invasion on 24 February.
Russia claims to have seized strategic port city
Efforts underway to speed up border crossings
More than half of those who have fled Ukraine have crossed into neighbouring Poland. Hungary, Moldova, Hungary and Slovakia have also welcomed many refugees.
Around 575,100 people have entered Poland from Ukraine since Russia launched its invasion of the country, the Polish border guard has said.
The Border Guard said that around 95,000 people entered the country yesterday, with 27,100 more people crossing the border by 6am this morning.
The UNHCR said more than 505,500 people have left Ukraine for Poland in the past week.
Mr Grandi said UNHCR staff and other humanitarians were "working where and when they can in frightening conditions" inside Ukraine.
"Our staff stay, even at great risk, because we know the needs in the country are huge."
The UN refugee chief also hailed the "remarkable" response of governments and local communities in surrounding countries that have received the more than one million refugees.
"International solidarity has been heart-warming," he said.
"But nothing - nothing - can replace the need for the guns to be silenced; for dialogue and diplomacy to succeed. Peace is the only way to halt this tragedy."
EU agrees to offer protection to Ukraine refugees
The EU has agreed to approve temporary protection for refugees fleeing the war in Ukraine, while also setting up a humanitarian hub in Romania.
The moves by the European Union came in parallel to its sanctions on Russia imposed in successive waves over the course of the invasion.
In Washington, the administration of President Joe Biden announced a similar move, granting temporary protected status to Ukrainians already in the country.
That means they can remain in the US and the threat of deportation is lifted.
EU interior ministers agreed in a Brussels meeting to activate a temporary protection mechanism drawn up two decades ago - in response to the wars in the former Yugoslavia - but never used.
European home affairs commissioner Ylva Johansson, who attended the meeting, tweeted that it was a "historic decision".
French Interior Minister Gerald Darmanin said: "The European Union will accord temporary protection to all those fleeing the war in Ukraine."
Both later told a media conference that the decision was unanimously taken by the ministers representing the EU's 27 nations.
Mr Darmanin said the temporary protection would apply to Ukrainians and their family members crossing into the EU, as well as anyone who had prior refugee status in Ukraine.
Ms Johansson said there were many foreigners, including students, who had been in Ukraine when the Russian assault started, and stated: "They are not covered by Temporary Protection directives, but they are being helped out of Ukraine."
"The significance of this moment for Europe cannot be underestimated," the Oxfam charity said in a statement of the EU's decision, calling it "a turning point for Europe".
It said the EU protection mechanism "offers a direct lifeline to people fleeing from danger in Ukraine" and now "all EU member states need to chip in and take responsibility".
With political agreement reached, it was expected to come into force within days, once the EU legislation is finalised.
