As Russian air strikes hit overnight, many packed up and left Kyiv for the countryside or the border with the European Union.
Traffic jams clogged the roads out of some cities, and some walked on foot into Poland and Hungary.
The UN refugee chief warned that the humanitarian impact will be "devastating".
"We are particularly worried about displacement - about people on the move," Filippo Grandi told the BBC's Lyse Doucet, who is in Kyiv.