The cross-border incursion appears to be Ukraine's most serious since Russia's invasion began in February 2022, and may involve up to 1,000 troops, according to Moscow's estimates.
"To be honest, I feel proud of our guys and happy," said Tetiana Krapyvka, a financier in the Ukrainian capital.
"I believe that it was a prepared operation and maybe personnel reserves were used, but it was prepared well," she added.
Ukraine has not officially taken responsibility for the operation, which began on Tuesday morning, but officials have blamed Moscow for inciting the attack.
Ukraine's President Volodymyr Zelensky said that "Russia brought the war to our land and should feel what it has done".
Russian President Vladimir Putin has called it a "large-scale provocation" by Kyiv and Russia's top general vowed on Wednesday to crush the incursion.
"With these actions, Ukraine is strengthening its defence capabilities and its ability to defend itself," said Anastasia Volkova, a cultural manager in the Ukrainian capital
"This is a success of our military, including in the most difficult areas of the front line."
The independent US-based Institute for the Study of War (ISW) said Ukraine had made significant territorial gains in the first two days of the incursion.
The Russian defence ministry said its troops were "continuing to destroy" armed Ukrainian units and were using air strikes, rocket and artillery fire to try to push them back.
- Show them 'what war is' -
Many residents praised the attack as payback for Russia's invasion, now in its third year.
A local resident in Kyiv, Rita Simon, told AFP that the Kursk region operation was "necessary" to show Russia the consequences of war.
"They came to our land, and now let them feel what our mothers, our boys who are fighting now, feel," she said.
"Let them feel what it's like to live and lose their children, and let their children feel what we feel."
Volodymyr Borodyansky, a worker in the tourism sector, had the same sentiment.
"Let those in Moscow also know what war is," he said.
Another resident, 27-year-old Volodymyr Pyatov, said the operation would give Ukraine the upper hand in the war.
"It seems to me that the military leadership knows better what is best for bringing victory closer," he said.
"I hope that this will somehow help to withdraw enemy forces from other fronts and make it easier for the Ukrainian military on the eastern and southern fronts," he added.
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