Russia aims to "deprive people of light and heat" by causing power cuts across eastern Ukraine in revenge for a Ukrainian counter-offensive, President Volodymyr Zelensky has said.
The blackouts have reportedly affected around nine million people in eastern regions including Kharkiv and Donetsk.
It comes after Ukraine said it had retaken over 3,000 sq km (1,158 sq miles) during a rapid counter-offensive in the east.
The BBC cannot verify these figures.
Kharkiv mayor Ihor Terekhov said Russian attacks on civilian infrastructure left much of his city without power or water.
He called it a vile and cynical attempt at revenge for the Ukrainian army's recent successes.
What sounded like two further missile strikes were heard later in the evening, the BBC's international correspondent Orla Guerin reported from Kharkiv.
Mr Terekhov and the region's governor called for calm, saying the emergency services were working to repair damage and put out fires.
The governor of the neighbouring Sumy region said over 130 settlements in one district alone were without power.
Similar problems have been reported in Dnipropetrovsk and Poltava regions.
In a defiant post on social media after the power cuts, President Zelensky said standing up to Russia was more important to Ukraine than suffering cold, hunger, darkness or thirst.
It comes after a remarkable Ukrainian advance, which if confirmed will mean Ukrainian forces have tripled their territorial gains in little over 48 hours.
President Zelensky said 1,000 sq km had been retaken on Thursday evening. That figure rose to 2,000 sq km on Saturday evening, then to 3,000 on Sunday.
Journalists have been denied access to the front lines but several videos on social media show Ukrainian troops present in towns and villages that were until recently held by Russia.