Explosions were heard in Kiev and other Ukrainian cities, a day after Russia accused Ukraine of carrying out a drone attack on the Kremlin.
Attacks were also reported in Zaporizhzhia and Odesa in the south.
The airstrikes come as Ukrainian President Volodymyr Zelenskyy is scheduled to speak in The Hague as part of a surprise visit to the Netherlands.
He will also visit the International Criminal Court, which is investigating alleged Russian war crimes in Ukraine.
Russia has accused Ukraine of attempting to assassinate President Vladimir Putin, but Mr Zelensky has denied that his country carried out the attack.
On Wednesday he said: “We do not attack Putin or Moscow. We fight on our land. We are defending our villages and cities.”
Mr Zielinski was speaking in Finland, where he made a surprise visit and met with his Finnish counterpart Sauli Niensto and the leaders of Sweden, Norway, Denmark and Iceland.
During his visit to the Netherlands, Mr. Zielinski is expected to meet Dutch Prime Minister Mark Rutte. Dutch media say the two are likely to discuss Mr Zielinski’s demands for more military cooperation – namely long-range weapons and fighter jets.
Last January, Mr Rutte said the delivery of fighter jets was not prohibited, although such a move would be “a really big next step”.
After the alleged drone strikes, Russia threatened to retaliate when and where deemed necessary.
On Wednesday, 21 people were killed in Russian attacks on the southern Kherson region of Ukraine. Officials said the dead included supermarket customers and employees of an energy company who were doing repair work.
And early Thursday morning, air raid sirens went off in many Ukrainian regions. Loud explosions were reported in Kiev and Odessa.
At the same time, a drone struck an oil refinery in southern Russia, setting part of it on fire – the latest in a series of explosions, fires and drone attacks in Russia in recent weeks.
Some observers have argued that the alleged drone attack on the Kremlin was orchestrated and deliberate by Russia.
The Institute for the Study of War said it is “highly unlikely that two drones could have penetrated multiple layers of air defenses and been detonated or shot down in such a way over the very heart of the Kremlin.” “Which is a well-done in-camera image capture.”
It said “Russia may have carried out the attack in an attempt to bring the war home to a Russian domestic audience and create the conditions for wider social mobilization”.
Ukrainian Air Force spokesman Yuriy Ikhnat said he believed Russia had launched the attack on the Kremlin to “show some kind of tension on the part of Ukraine.”
But other observers disagreed, saying Russia would have no interest in showing itself “weak” by launching an attack that would make the Kremlin look weak.
It will also raise questions about how safe Mr Putin is – and about the effectiveness of Russian air defences.
