The attack on Monday night hit an educational institution, residences, an underground station and medical facilities.
Kyiv Mayor Vitali Klitschko, writing on Telegram, said one person had been killed near a supermarket. Ukraine's state rescue services said a body was found by a heavily damaged apartment building.
Klitschko put the number of injured at 10, many of them outside a student dormitory at the State Trade and Economics University on the east side of the Dnipro River. Nine people were being treated in hospital, with three in serious condition.
President Volodymyr Zelenskiy gave no precise figures for the dead and injured.
"For the moment, the number of people is not exactly known," Zelenskiy said on X.
"But, unfortunately, there are casualties. There are injured. Rescue operations are continuing."
He repeated his call to eliminate delays in securing air defence arms quickly to counter Russian air attacks.
"It is precisely in such conditions, when the threat is constant, when strikes can happen at any moment and on cities and villages, that artificial delays in air defence for Ukraine are helping terror," he said.
Earlier in the day, Zelenskiy said Ukraine had been waiting for US Defence Secretary Pete Hegseth to sign off on critical missile supplies.
"In particular, we have been waiting for nearly a month for Mr Hegseth to sign off on missiles for Patriots and NASAMS. I instructed our team to raise this with the American side once again," the president said on X.
Unofficial Telegram channels posted pictures of damage to an underground station and a tram.
Klitschko said explosions shattered windows at two medical facilities. He said two members of an emergency crew had been injured after rushing to the scene of one strike.
Reuters witnesses said a series of explosions jolted the city. They said power had been cut in districts on the east side of the Dnipro River.
Moscow has long denied that it targets civilians.
with DPA
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