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.
"We estimate that more than 100,000 must have already moved out of their homes in Ukraine seeking safety in other parts of the country," he said, speaking from Geneva. Mr Grandi added that his agency had stockpiled supplies to provide aid to people in Ukraine - but it could not act while military action was still happening.
Ukraine's European neighbours say they are ready to receive refugees fleeing the country of 44 million people - one of the most populous in Europe.