There is no electricity in the second-largest city of Kharkiv, says regional head Oleg Synehubov.
Fifteen blasts were reported in Kharkiv, while more than 53,000 households in Odesa were without power.
Ukraine’s energy minister, German Galushchenko, accused Russia of trying to provoke “a large-scale failure of the country’s energy system”.
A power line feeding the Zaporizhzhia nuclear plant had been cut, he added.
Regional head Ivan Fedorov said the power station was “on the verge of a blackout”, adding that seven buildings in the region had been destroyed and 35 others damaged.
President Volodymyr Zelensky said Russia had launched more than 60 Shahed drones and about 90 missiles into Ukraine during the wave of overnight attacks.
At least two people have been reported killed and 14 wounded.
The fresh onslaught comes a day after Russian forces launched one of their biggest air strikes in weeks on the Ukrainian capital, Kyiv. At least 17 people, including a child, were injured by falling debris.
The latest attack prompted President Zelensky to renew his call for more military aid and additional air defence systems from Western allies.
He said that the shortfall in ammunition facing his troops was “humiliating” for Europe, adding: “Europe can provide more, and it is crucial to prove it now.”
Ukraine’s state hydropower company said a Russian strike had also hit its largest dam, the DniproHES in Zaporizhzhia.
Video footage appeared to show the dam on fire, but there seems to be no threat of imminent breach.
Social media videos showed a trolleybus with passengers, which was crossing the dam when it caught fire after a missile strike.
Police in Zaporizhzhia say traffic across the dam has been completely halted.
Blasts were also reported from Kryvyi Rih, which is President Zelensky’s hometown, and Vinnytsia, both in central Ukraine. They damaged a “critical infrastructure object”, said Ukrainian officials.
Attacks on energy grids have been part of Russia’s warfare against Ukraine. In the winter of 2022, Moscow carried out widespread strikes on Ukrainian power infrastructure, plunging millions of people into darkness and depriving them of heat, power and water. The attacks left 17 million Ukrainians without a regular supply of electricity for extended periods.
But the head of the Ukrainian grid operator, Volodymyr Kudrytsky, said they were not nearly as bad as the attacks on Thursday night.
He said: “Dozens of grid facilities have been hit. This is on a global scale.”
Mr Kudrytsky added that the worst affected was Kharkiv, where “Russia literally tried to destroy all the main energy facilities feeding the city”.
President Zelensky has often described the Russian attacks on power infrastructure as “energy terrorism”.
Russia’s full-scale invasion of Ukraine has now entered its third year, and there are no signs that the biggest war in Europe since World War Two could end any time soon.