Ukrainian public broadcaster Suspilne quoted emergency services as saying there were dead and wounded in Kyiv but gave no further details of casualties.
"Several explosions in the Shevchenskivskyi district - in the centre of the capital," Kyiv Mayor Vitali Klitschko Klitschko said on the Telegram messaging app. "Details later."
Reuters witness saw a huge carter at one of the city centre's intersections and nearby cars completely wrecked, blackened and pitted with shrapnel.
Explosions were also reported in Lviv, Ternopil and Zhytomyr in Ukraine's west, and in Dnipro in central Ukraine.
Russia abandoned an early advance on Kyiv in the face of fierce resistance bolstered by Western arms.
Since then Moscow and its proxies have focused on the south and Donbas, an eastern territory made up of Luhansk and its neighbour Donetsk, deploying overwhelming artillery in some of the heaviest ground fighting in Europe since World War Two.