Sites have also been targeted in Khmelnitskyi in western Ukraine, and the central Dnipropetrovsk region.
While Ukrainian air defences said they had shot down a number of missiles and drones, it was reported Russia had also fired S-300 rockets, which cannot be intercepted.
Zaporizhzhia city council secretary Anatolii Kurtiev said the city had been hit 17 times in one hour, which he said made it the most intense bombardment since the beginning of the invasion last February.
In Kharkiv, authorities were trying to establish the number of casualties and scale of the destruction, with mayor Ihor Terekhov saying there may be disruptions to heating and the electricity and water supply.
Serhiy Popko, head of Kyiv city military administration, told residents: “There is a big threat of missile attack. I want to stress again – do not ignore the air alert sirens.”
Military analysts say Russian President Vladimir Putin is hoping Europe’s support for Ukraine will ebb, as Moscow is believed to have launched its expected major offensive in a bid to make territorial gains as the anniversary of the invasion approaches.
Kyiv’s military intelligence agency said Kremlin forces have intensified attacks in the partially occupied Donetsk and Luhansk regions, with the aim to grab full control of the entire industrial region, known as the Donbas.
Moscow-backed separatists have been fighting Ukrainian forces there since 2014.
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23:16
Zelenskyy speaks in Westminster Hall
Luhansk governor Serhiy Haidai told Ukrainian radio: “Over the past week to 10 days, the frequency of shelling has increased.
“The daily number of attacks has increased.”
Meanwhile, Ukrainian President Volodymyr Zelenskyyused a visit to the UK and Europe to urge the supply of advanced fighter jets, sparking a warming by Moscow.