Ukrainian troops have advanced up to 30km inside Russia, in what has become the the deepest and most significant incursion since Moscow began its full-scale invasion of Ukraine in February 2022.
Russia's defence ministry said its forces had engaged Ukrainian troops near the villages of Tolpino and Obshchy Kolodez, as the offensive in the Kursk region entered a sixth day.
Foreign ministry spokesperson Maria Zakharova accused Kyiv of "intimidating the peaceful population of Russia".
Overnight, President Volodymyr Zelensky directly acknowledged the attack for the first time, saying Ukraine was pushing the war to "the aggressor's territory".
"Ukraine is proving that it can indeed restore justice and ensure the necessary pressure on the aggressor," Mr Zelensky told the country in his nightly address from Kyiv.
He went on to thank Ukraine's "warriors" and said he had discussed the operation in Russia with the country's top military commander - Gen Oleksandr Syrskyi.
A senior Ukrainian official told the AFP news agency that thousands of troops were engaged in the operation, far more than the small incursion initially reported by Russian border guards.
While Ukrainian-backed sabotage groups have launched intermittent cross-border incursions, the Kursk offensive marks the biggest co-ordinated attack on Russian territory by Kyiv's conventional forces.
“We are on the offensive. The aim is to stretch the positions of the enemy, to inflict maximum losses and to destabilise the situation in Russia as they are unable to protect their own border,” the official said.
Russia's defence ministry said on Sunday that its forces had "foiled attempts by enemy mobile groups with armoured vehicles to break through deep into Russian territory".
But in an apparent admission that Kyiv's forces have now advanced deep into the Kursk border region, the defence ministry reported engaging Ukrainian forces near the villages of Tolpino and Obshchy Kolodez - which are about 25km and 30km from the Russia-Ukraine border.
Ukrainian troops have claimed to have captured a number of settlements in the Kursk region. In Guevo, a village about 3km inside Russia, soldiers filmed themselves removing the Russian flag from an administrative building.
Clips have also emerged of Ukrainian troops seizing administrative buildings in Sverdlikovo and Poroz, while intense fighting has been reported in Sudzha - a town of about 5,000 people.
Ukrainian troops have already filmed themselves outside Sudzha at a major gas facility involved in the transit of natural gas from Russia to the EU via Ukraine, which has continued despite the war.
In Sumy, which borders the Kursk region, BBC reporters witnessed a steady stream of armoured personnel carriers and tanks moving towards Russia.
The armoured convoys are sporting white triangular insignias, seemingly to distinguish them from hardware used within Ukraine itself. Meanwhile, aerial photos have appeared to show Ukrainian tanks engaged in combat inside Russia.
Photos analysed by BBC Verify also appeared to show Russia constructing new defensive lines near the Kursk nuclear power plant. Ukrainian forces are said to have advanced within 50km (31 miles) of the facility.
Contrasting satellite imagery of the same location captured yesterday with imagery from a few days earlier, images show several newly constructed trench lines in the vicinity, with the nearest roughly 8km (5 miles) from the plant.
