One of the drone strikes, targeting an airfield far from Ukraine's borders, destroyed military transport planes on the ground.
In a separate incident, Ukraine's military said six of its servicemen had died on Tuesday when two helicopters crashed near Bakhmut in the country's east.
It gave no details of what happened but said all the men were officers.
Late on Wednesday, Russia said its anti-aircraft units had downed two Ukrainian drones in its southern Bryansk region.
The head of Crimea, annexed by Russia from Ukraine in 2014, said a Ukrainian cruise missile had been intercepted there.
The previous night, Russian officials described attacks on Bryansk and five other regions, including Moscow and Pskov in the north, where four military transport planes were damaged in a huge blaze at an airfield.
The Russian foreign ministry said the attacks would "not go unpunished" and the drones could not have reached so far into Russian territory without Western help.
The attacks coincided with Russia's most severe air strikes on Ukraine's capital for months.
Authorities in Kyiv reported at least two people had been killed as debris from intercepted missiles fell in four locations.
Reuters captured footage of a fireball falling out of the night sky close to a supermarket.
Moscow said it hit command and intelligence targets.
In Russia, the governor of Pskov posted a video on Telegram showing a huge fire with the sounds of sirens and an explosion at the air base.
Another video posted online showed anti-aircraft systems in action around the city, 32km east of Russia's border with NATO member Estonia.
Moscow said it had thwarted all the attacks.
Russia typically describes all Ukrainian drone strikes as unsuccessful, regardless of the damage on the ground.
Kremlin spokesman Dmitry Peskov said Russia was working out where the drones were launched from to prevent further strikes.
President Vladimir Putin had been informed immediately, as would be the case in any such "massive attacks", Peskov said.
The Kyiv government confirmed the Russian planes had been destroyed in Pskov.
It generally withholds comment on strikes on territory inside Russia although it says it has a right to hit military targets.
"Yes, four IL-76 transport planes were destroyed in Pskov at an airfield," Andriy Yusov, a spokesman for Ukraine's GUR military agency, told Reuters.
"They are beyond repair.
"Also, several other of those (aircraft) are damaged, but the information is being checked."
Attacks on Russia in recent weeks, including repeated drone strikes on central Moscow, have brought the war home to many Russians for the first time.
"So long as Putin remains president, the war will continue pulling Russia deeper and deeper into the abyss of chaos," Mykhailo Poldolyak, a senior aide to Ukrainian President Volodymyr Zelenskiy, wrote on social media platform X.
Moscow has relentlessly pounded Ukrainian cities with long-range missiles and drone strikes throughout the war.
Thousands of Ukrainian civilians have been killed.
Ukraine said its air defences had shot down 28 Russian missiles and 15 out of 16 drones fired overnight.
Ukrainian forces have been conducting an offensive for almost three months but have yet to break through Russia's mined and fortified defences, although they said this week they had penetrated the first main defensive line.
Military spokesman Andriy Kovalyov told national television on Thursday Ukrainian forces were making progress in the east near Bakhmut, and in the Melitopol sector, where they are trying to move southward to the Sea of Azov.
Russian military accounts spoke of forces repelling five attempted Ukrainian advances in the east.
