The attacks are aimed at Ukraine's capital Kyiv and Kharkiv in eastern parts of the country.
Kyiv Mayor Vitali Klitschko said on Telegram that one person had been killed and several others injured in the city. Fires broke out in several places after drones struck various buildings, according to the mayor.
The attacks come after representatives from Ukraine, Russia and the United States held talks in Abu Dhabi, the capital of the United Arab Emirates, on Friday.
The talks are set to resume on Saturday, according to Ukrainian President Volodymyr Zelenskyy.
