Kyiv Mayor Vitali Klitchko said multiple blasts hit a central district in the Ukrainian capital on Monday.
"Several explosions in the Shevchenkivskyi district — in the center of the capital. All services are going to the places. Details later," he wrote on Telegram.
The district houses a university, several government offices and includes the historic old town.
The blasts come after months of relative calm in the capital, which has been spared most of the fighting that has raged in the country's east.
A spokesperson for the Emergency Service in Kyiv told several media outlets there were an unspecified number of deaths and injuries, with rescuers at work in several locations.
Meanwhile, authorities reported explosions in the western city of Lviv and the central city of Dnipro.
The governor of Lviv said strikes had targeted "energy infrastructure" facilities near the city.
The Ukrainian presidency said there were explosions in "many" other Ukrainian cities Monday morning.
"Ukraine is under missile attack," Kyrylo Tymoshenko, deputy head of the president's office, said on social media, advising people to take shelter.
Ukrainian President Volodymyr Zelenskyy wrote on Telegram Monday that there were dead in wounded in explosions across the country, adding that Russia is trying to wipe Ukraine "off the face of the earth."
Putin calls bridge blast 'act of terrorism'
Russian President Vladimir Putin said the blast on Kerch Bridge was a terrorist attack orchestrated by Ukraine. He is to hold a meeting of his security council on Monday.
"There is no doubt. This is an act of terrorism aimed at destroying critically important civilian infrastructure. This was devised, carried out and ordered by the Ukrainian special services," Putin said in a video on the Kremlin's Telegram channel.
Security Council Deputy Chairman Dmitry Medvedev said to TASS news agency, "Russia can only respond to this crime by directly killing terrorists, as is the custom elsewhere in the world. This is what Russian citizens expect."
The Kerch Bridge is a key link between Russia and Crimea, built after Moscow illegally annexed the territory in 2014. It is also an artery for the port of Sevastopol, where the Russian Black Sea fleet is based.
Putin met Sunday with Alexander Bastrykin, the head of the Investigative Committee, who said the explosion on the bridge was caused by a vehicle that had traveled through Bulgaria, Georgia, Armenia, North Ossetia and Russia's Krasnodar region.
Ukrainian special services were helped by Russians as well as other foreign nationals, he said in a video posted on Telegram.
