Groups of Ukrainian refugees walk along the road between Lviv and Shehyni, in Volytsya, Ukraine, Saturday March 5, 2022. AP
UNHCR, the UN's refugee agency, recorded 2,155,271 refugees on its dedicated website -- 143,959 more than the previous count on Tuesday.
"Behind the monolithic statistics are two million stories of separation, anguish, and loss," UNHCR chief Filippo Grandi said.
Families have been "senselessly ripped apart", plunged into "despair and unimaginable suffering" by the "brutal war", he said.
Authorities and the UN expect the flow to intensify as the Russian army advances deeper into Ukraine, particularly as it approaches the capital, Kyiv.
Before Russia invaded, more than 37 million people lived in Ukrainian territory under the control of the central government.
Besides those who have left, an unknown number have been displaced from their homes within the country.
Here is a breakdown of where refugees from Ukraine are, according to the UN Refugee Agency:
More than half of those who have fled Ukraine are now in Poland, with the UNHCR saying 1,294,903 refugees are now in the country.
Poland has championed the cause of Ukrainian refugees. The government has set up reception centres and charities have mobilised in a massive aid effort, helped by the estimated 1.5 million Ukrainians already living in the EU member state.
Elsewhere in Europe
Some 235,745 people who have fled Ukraine have now moved beyond neighbouring states to other European countries, according to the UNHCR.
Some 203,222 refugees are now in Hungary -- nearly 10 percent of the total who have fled Ukraine. The number was up 11,874 on Tuesday's figure.
The country has five border crossings with Ukraine and several border towns, including Zahony, have turned public buildings into relief centres, where Hungarian civilians are offering food or assistance.
Across Ukraine's shortest border, some 153,303 refugees are now in Slovakia.
The UNHCR says the number of refugees who have crossed Ukraine's longest border into Russia since the invasion is 99,300.
An additional 96,000 people moved to Russia from the separatist eastern Donetsk and Lugansk regions between February 18 and 23, in the days before the Russian invasion, according to the UN refugee agency.
Some 85,444 refugees from Ukraine are now in Romania. Two camps have been set up, one in Sighetu Marmatiei and the other in Siret.
Some 82,762 refugees were now in Moldova, according to figures updated to the end of Sunday, though many thousands more have passed through the non-EU state on their way to other countries.
By the end of Sunday, Moldova had welcomed 259,000 refugees coming from Ukraine, including 230,000 Ukrainians.
"Over 147,000 Ukrainian refugees have since then proceeded to Romania," UNHCR said.
Some 592 refugees had made it to Belarus, according to UNCHR