President Volodymyr Zelensky said Ukraine was invading Russian territory to "restore justice" and pressure Moscow's forces.
Russian authorities rushed to evacuate residents on Saturday. They imposed a broad security regime in three border regions. At the same time, Belarus, a staunch ally of Moscow, sent more troops to its border with Ukraine, accusing Kyiv of violating its airspace.
In his video message, Zelensky said he discussed the operation with Ukraine's top commander, Alexander Sirsky, vowing to respond in kind after Russia launched its full-scale invasion of its neighbor in February 2022.
"Today, I received several reports from Commander-in-Chief Sirsky on the front line about our actions to push the war into the aggressor's territory," he said.
"Ukraine is proving that it can restore justice and provides the exact type of pressure needed on the aggressor."
The Russian Ministry of Defense reported that 14 Ukrainian drones and 4 Tochka-U tactical ballistic missiles were destroyed overnight over the Kursk region, and 18 drones over other areas of Russia, which Ukraine frequently attacks.