Several fires have broken out in buildings and cars around the capital, reports Klytjko on social media.
Nearly 40 people have been taken to hospital, including six children. Around 15 more people have received medical attention on site.
A particularly affected area is in western Kyiv, where a multi-family house has been hit and started burning.
"The house is destroyed, people are trapped under the rubble. Rescue personnel and medical staff are working on site", writes the mayor, who has urged people to seek shelter.
Ukraine's air force has issued a nationwide air raid alert and detected a large number of drones and rockets in Ukrainian airspace.
Residential buildings in the Kyiv region outside the capital have also been damaged, according to local authorities.
Powerful explosions have also been heard in Kharkiv in the northeast, Ukraine's second-largest city. At least two people have been injured and both rockets and drones have struck in several rounds, according to the city's mayor.
Explosions have also been heard in Pavlohrad in Dnipropetrovsk, according to the governor there.
