The overnight attacks follow Russia's biggest drone strike on Ukraine on Monday - part of intensified operations that Moscow said were retaliatory measures for Kyiv's recent brazen attacks inside Russia.
Medics were called to four districts of Kyiv after midnight on Tuesday, including the historic Podil neighbourhood, Mayor Vitali Klitschko said on the Telegram messaging app.
The military said the strikes were still ongoing and urged people to seek bomb shelters.
"Enemy drones are simultaneously attacking several districts of the city," Timur Tkachenko, head of Kyiv's military administration said on Telegram. "There is damage to residential buildings and fires. Rescuers are working at the sites."
No casualties have been reported from the maternity hospital, with patients and staff evacuated.
Earlier, Russia and Ukraine exchanged prisoners of war under the age of 25 in emotional homecoming scenes, the first step in a series of planned prisoner swaps that could become the biggest of the war so far.
The exchange was the result of direct talks between the two sides in Istanbul on June 2 that resulted in an agreement to exchange at least 1200 POWs on each side and to repatriate thousands of bodies of those killed in Russia's war in Ukraine.
The return of POWs and the repatriation of the bodies of the dead is one of the few things the two sides have managed to agree on as broader negotiations have failed to get close to ending the war, now in its fourth year.
Fighting has raged on, with Russia saying on Monday its forces had taken control of more territory in Ukraine's east-central region of Dnipropetrovsk and Ukraine saying Russia had launched its largest drone attack of the war.
Officials in Kyiv said some of the Ukrainian prisoners who came home on Monday had been in Russian captivity since the beginning of the war.
At a rendezvous point for the returning Ukrainian prisoners, soon after they crossed back into northern Ukraine, an official handed one of the freed men a mobile phone so that he could call his mother, a video released by Ukrainian authorities showed.
The released Ukrainian men were later taken by bus to a hospital in northern Ukraine where they were to have medical checks and be given showers, food and care packages including mobile phones and shoes.
Jubilation was tinged with sadness because outside the hospital were crowds of people, mostly women, looking for relatives who went missing while fighting for Ukraine.
The women held up pictures of the missing men in the hope that one of the returning POWs would recognise them and share details about what happened to them.
Some hoped their loved ones would be among those released.
Neither side said how many prisoners had been swapped on Monday but the Russian Defence Ministry said in its own statement that the same number of military personnel had been exchanged on each side.
Ukrainian President Volodymyr Zelenskiy, speaking in his nightly video address, said the exchanges would be conducted over several stages.
Zelenskiy also said Ukrainian forces were engaged in heavy fighting near Pokrovsk in the east and inside Russia's Kursk region and also in Ukraine's Sumy region, where Russian forces are trying to establish a presence in several border villages.
The Russian military said its returned servicemen were now in Belarus, where they were receiving psychological and medical assistance before being transferred to Russia for further care.
Footage broadcast by Russia's RIA state news agency showed a group of freed Russian soldiers on board a coach raising their hands in the air and shouting: "Hurrah we're home."
"It is very difficult to convey what I'm feeling inside now. But I am very happy, proud and grateful to everyone who took part in this process, in the exchange and bringing us home," one freed Russian soldier said.
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