At least three people - including a 14-year-old girl - were taken to hospital after explosions hit Kyiv, while blasts were also heard in the cities of Kharkiv, Odesa, Lviv and Zhytomyr.
Mykhailo Podolyak, an adviser to Ukrainian President Volodymyr Zelensky, said Russia launched more than 120 missiles.
In Kyiv, air defence systems were activated to fend off the ongoing missile attack, according to the regional administration, while Ukrainian authorities in several regions said some incoming Russian missiles were intercepted.
The governor of southern Ukraine's Mykolaiv province, Vitaliy Kim, said five missiles were shot down over the Black Sea.
Kyiv mayor Vitali Klitschko warned on Thursday that there could be power cuts in the capital, asking people to stockpile water and to charge their electronic devices.
As the latest wave of Russian strikes began, authorities in the Dnipro, Odesa and Kryvyi Rih regions said they switched off electricity to minimise the damage to critical infrastructure facilities if they were hit.
In a village near the city of Ivano-Frankivsk, officials said an unexploded missile had landed in a house.
Earlier this month, the US agreed to give a Patriot missile battery to Ukraine to boost the country's defence.
The US and other allies also pledged to provide energy-related equipment to help Ukraine withstand the attacks on its infrastructure.
Mr Podolyak said that Russia was aiming to "destroy critical infrastructure and kill civilians en masse".
"We're waiting for further proposals from 'peacekeepers' about 'peaceful settlement,' 'security guarantees for RF' and undesirability of provocations," he wrote on Twitter, a sarcastic reference to statements from some in the West who urged Ukraine to seek a political settlement of the conflict.
Russian president Vladimir Putin has insisted that he has not targeted civilians in the conflict. However, he recently admitted that he has targeted infrastructure such as energy supplies, which prompted some international leaders including French president Emmanuel Macron to warn that the move could be regarded as a war crime.