At least three people – including a 14-year-old girl – were taken to hospital after explosions hit the capital Kyiv, Mayor Vitaliy Klitschko said.
Blasts were also heard in the cities of Kharkiv, Odesa, Lviv and Zhytomyr.
The regional leader of the southern province of Odesa, Maksym Marchenko, spoke of a “massive missile attack on Ukraine”.
The Ukrainian Air Force said Russia was attacking the country from “various directions with air and sea-based cruise missiles”. It added that a number of Kamikaze drones had also been used.
Air raid alerts sounded in all regions of the country on Thursday morning. Presidential adviser Oleksiy Arestovych urged civilians to seek shelter and said the country’s air defences were operating.
In Kyiv, two homes were damaged by debris from an intercepted missiles, according to the city military administration. Mr Klitschko said 16 missiles were destroyed over the city by air defences.
In the southern region of Mykolaiv, Governor Vitaly Kim wrote that five missiles were intercepted by air defences.
Mr Marchenko said 21 missiles were shot down in the Odesa region. He added that missile fragments had hit a residential building but no casualties were reported.
And in the Western city of Lviv Mayor Andriy Sadovy said several explosions had been reported.
Mr Podolyak accused Moscow of seeking “to destroy critical infrastructure and kill civilians en masse”.
