Moscow carried out the strikes as trilateral, US-brokered talks between Russia and Ukraine continued into a second day in the United Arab Emirates.
Ukrainian President Volodymyr Zelenskiy says it was the heaviest attack in weeks.
"Russia launched a massive attack on our regions during the night, firing 370 combat drones and 21 missiles of various types," Zelenskiy said on X on Saturday.
"Every such Russian strike on our energy infrastructure proves that there must be no delays in supplying air defences," he added. "No blind eye can be turned to these strikes."
In addition to Kiev and the surrounding area, the regions of Kharkiv, Sumy and Chernihiv, located near the border with Russia, were also struck.
Zelenskiy reported one fatality and dozens of injuries, as well as further severe damage to the energy supply infrastructure.
Deputy Prime Minister Oleksiy Kuleba said more than 3200 buildings in the capital remained without heating in the late evening, down from 6000 in the morning. Night-time temperatures were hovering around -10 degrees Celsius
The Ukrainian air force and The Kyiv Independent news outlet said shortly after midnight that the city was under heavy attack. Drones and missiles struck multiple districts, officials said, leaving at least one person dead and four others injured.
In the eastern Ukrainian city of Kharkiv, close to the Russian border, Mayor Ihor Terekhov said at least 19 people were injured.
Several hospitals and residential buildings were damaged, while some people were trapped under the rubble.
The governor of Kharkiv region, Oleh Syniehubov, said a pregnant woman and a child were among the injured.
Meanwhile Russian officials said the western Russian city of Belgorod, near the border with Ukraine, was hit by a heavy rocket attack.
"According to our information, it was the most massive rocket attack on Belgorod (since the start of the war)," wrote the governor of the region, Vyacheslav Gladkov, on Telegram.
He added that energy facilities had been hit. However, according to initial information, there were no deaths or injuries.
Russia has been systematically shelling the infrastructure of neighbouring Ukraine since the start of the war. The energy grid there has now been weakened to such an extent that in many places people have to cope without electricity and heating for hours every day.
But the Russians themselves are also increasingly feeling the consequences of the war.
Belgorod is one of the most affected regions in Russia. Due to Ukrainian attacks, the electricity supply there is also restricted.
The attacks came as Ukrainian and Russian representatives have been holding talks in Abu Dhabi since Friday, mediated by the United States, aimed at ending Russia's war against Ukraine.
The negotiations in the capital of the United Arab Emirates on Friday and Saturday mark the first direct talks between the two sides in months.
The delegations returned to their hotels after meeting on Saturday and there were no plans to continue, the Russian state news agency TASS cited sources close to the negotiations as saying. No concrete results have been announced so far.
The negotiations are aimed at finding a way to end the war against Ukraine ordered by Russian President Vladimir Putin almost four years ago in February 2022.
with Reuters
