More than three million people, the vast majority women and children, have fled Ukraine since Russia invaded on February 24, the United Nations said.
Among those fleeing Ukraine are over one million children, while many more have been internally displaced.
United Nations chief Antonio Guterres warned the impact of the war on civilians was "reaching terrifying proportions”. The UN has called the exodus from Ukrainian cities under Russian attack as Europe's biggest refugee crisis since the second world war.
Shortly before dawn, large explosions were heard across Kyiv. A series of Russian strikes hit a residential neighbourhood in the capital, igniting a huge fire and a frantic rescue effort in a 15-story apartment building.
After days of relentless Russian shelling on encircled Mariupol, a convoy of 150-160 cars carrying hundreds of civilians managed to escape the besieged city on Monday. But hundreds of thousands of people remain trapped without heat, food or clean water as Russia renewed its offensive on the city.
The vast majority of those fleeing Ukraine are arriving in neighbouring Poland. New waves of exhausted refugees from Ukraine arrived by train at the Polish border town of Przemysl on Tuesday where they were welcomed by volunteers at the town's train station who provided them with food and other help.
Marina Solonenko, 35, reached Przemysl after travelling from Kharkiv, Ukraine's second-largest city. "Everything is bombed," she said. "The city is destroyed."
Others have been displaced in Ukraine, moving West ahead of the invading army. Pasha Bychkov, 10, said his family had escaped Kharkiv after a bomb struck their apartment building. They were now in Lviv, where he has resumed school.
Germany's Interior Ministry said so far 146,998 refugees from Ukraine had been registered coming to the country. He said the real number may differ if people did not register or had moved on to another country.
The UK has been criticised for being slow at offering Ukrainians help.
The Homes for Ukraine scheme, to be rolled out this week, will allow individuals, charities, community groups and businesses to bring people fleeing the war to safety - even if they have no ties to the UK.
Britons offering homes to Ukrainian refugees through a new humanitarian route will receive a “thank you” payment of £350 per month.
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