Russia shelled the northern city of Chernihiv all night, Ukraine's State Emergency Service said, BBC reported.
Blasts were also heard before dawn in the capital Kyiv, and second city Kharkiv, Ukrainian officials said.
Shells fell in Chernihiv all night, although only one injury has so far been reported in the city. The artillery assault began around 02.00, according to the Ukrainian State Emergency Service.
According to the agency, rockets struck a building housing a kindergarten, causing a fire. A shop in a central market was also struck, as well as a five-story residential apartment building. One woman was slightly injured, BBC reported.
More explosions were reported overnight in the Ukrainian capital and other cities around the country.
Ukraine claims that its troops managed to repel several attacks by Russian troops trying to storm the outskirts of the capital. "We showed that we can protect our home from uninvited guests," said Commander of the Armed Forces Colonel General Alexander Syrsky in a statement, BBC reported.
Explosions were also reported in Kharkiv in the east, which was the scene of fierce conflict on Sunday after Ukrainian forces recaptured the country's second-largest city from Russian forces.
President Zelensky says the next 24 hours - the fifth day of the Russian invasion - will be "crucial" for Ukraine, BBC reported.
The Russian rouble plunged to a new low against the dollar in the wake of severe sanctions, as analysts warn of a possible run on Russia's banks.
The EU is taking the unprecedented step of sending arms to Ukraine, the report said.