Air alerts sounded throughout the country during the night, and Klitschko reminded residents not to venture out.
"Stay in shelters!" the mayor wrote on his Telegram channel.
Klitschko said the wreckage of a unmanned aerial vehicle (UAV) fell in the south-eastern district of Darnytskyi.
Russia invaded neighbouring Ukraine more than a year ago and has repeatedly attacked it with combat drones since last autumn, often targeting public infrastructure such as power plants.
Russia's overnight air attack on Kiev - its eighth in May - was complex, with drones, cruise and likely ballistic missiles launched from different directions, Kyiv's officials said early on Tuesday.
"It was exceptional in its density - the maximum number of attack missiles in the shortest period of time," Serhiy Popko, head of Kyiv's city military administration, said in comments posted on the Telegram messaging app.
"According to preliminary information, the vast majority of enemy targets in the airspace of Kyiv were detected and destroyed!"
It was not immediately known how many objects were shot down over the city and if any of them managed to hit their target.
"According to preliminary information, the vast majority of enemy targets in the airspace of Kiev were detected and destroyed!"
It was not immediately known how many objects were shot down over the city and if any of them managed to hit their target.
After a weeks-long hiatus, Russia in late April resumed its tactic of long-range missile strikes and has launched a flurry of attacks in recent days, often targeting Kiev. Ukraine has been able to repel vast majority of the attacks so far.
On Tuesday, falling debris was reported in Kiev's Obolonskyi, Shevchenkivskyi, Solomyanskyi and Darnytskyi districts, officials said.
The damage in other districts was not significant and there was no immediate information on potential casualties there, the military administration said.
Air raid sirens blared across nearly all of Ukraine in the early hours of Tuesday, and were heard over Kiev and its region for more than three hours.
