The UN refugee agency says that about 660,000 people have fled Ukraine for neighbouring countries since the Russian invasion began.
The number, given on Tuesday, was up from a count of more than 500,000 a day earlier.
Shabia Mantoo, a spokeswoman for the UN High Commissioner for Refugees, said in Geneva that “at this rate, the situation looks set to become Europe’s largest refugee crisis this century”.
She said the agency is urging governments to continue allowing access to all those who are fleeing, including third-country nationals living in Ukraine who are forced to escape the violence.
She added: “We stress that there must be no discrimination against any person or group.”