The attacks struck residential buildings and civilian infrastructure.
In a suburb of the capital city, resident Inna Lukashenko said the strikes were relentless.
"We were under fire the whole night. First, there were drones. Then, we fell asleep for a second, and were woken up by explosions, hearing that missiles are in the air. Me and my child hid in the corridor. We were very scared."
In Kharkiv, emergency workers said they were working in four locations across the city and that most injuries were caused by shrapnel.
The attacks came hours after Russian President Vladimir Putin said an air attack on the Russian city of Belgorod would "not go unpunished".
Moscow said the strike on Saturday (December 30) killed at least 25 civilians and blamed it on Ukraine.
There was no official comment from Kyiv in the hours after the attack on Belgorod.
Kyiv mayor Vitali Klitschko said gas pipelines had been damaged in parts of the capital in Tuesday's assault and electricity and water had been cut off in several districts -
Bringing back memories of last winter when Russia pounded the energy grid with missiles, causing frequent power outages and plunging millions into darkness.
Ukrainian President Volodymyr Zelenskiy said at least 70 Russian missiles were shot down on Tuesday (January 2).
"I thank everyone in the world who helps, Patriots, IRIS, NASAMS; every such system has at least saved hundreds of lives. Russia will answer for every life taken away. Glory to Ukraine."
A Ukraine air force spokesperson said after the initial barrage, the situation remained tense and that air defenses were bracing for more Russian missile launches.
And Foreign Minister Dmytro Kuleba called on Western allies to accelerate supplies of air defense systems, long-range missiles and combat drones.
Russia has stepped up air strikes on Ukraine over the New Year period.
On Friday (December 29), it conducted the largest air attack since the start of the war.
