President Volodymyr Zelenskiy, speaking in his nightly video address on Tuesday, said one bomb struck a house, killing a woman. A second bomb hit the school.
"An ordinary school, where children happened to be at the time," he said.
"The building sustained considerable damage, but the children and teachers were in a shelter, and fortunately, thanks to that, no one was hurt at the school."
"Russian terrorists and simply unhinged scumbags. To strike at children, at schools, at ordinary homes like this. America, Europe, all our partners must not forget that Russia will not stop this madness without real pressure," he said.
"We expect steps in response to these strikes - and that they will go beyond lip service."
Governor Oleh Hryhorov, writing on Telegram, put the death toll at two, with three children among the injured. He said teachers and school officials had managed to evacuate 100 people into the shelter.
Sumy has been a frequent target of Russian attacks during its war with Ukraine. Russia's military has said it wants to establish a buffer zone in areas along the border.
Zelenskiy said Belgium would transfer three F-16 fighter jets to Ukraine by the end of the year to help in its war with Russia.
"There is an important agreement with Belgium on the transfer of three F-16 fighter jets by the end of this year," he wrote on X. "
We will work to receive these aircraft through an accelerated procedure."
Ukraine had already received "urgent deliveries" of F-16 ammunition from Belgium and Spain, among other countries.
Zelenskiy said later in his nightly video address that Ukraine had received delivery the day before of a "very good" assistance package of a NASAMS air defence system, developed jointly in the US and Norway.
Zelenskiy expressed thanks on social media to the US and Norway.
He also said that talks were proceeding to ensure that Ukraine has a supply of missiles for the Patriot air defence system, which play a critical role in intercepting Russian ballistic missiles.
Zelenskiy said tests were continuing to improve the efficiency of Ukrainian interceptors against Russian jet-powered drones, which fly higher and faster than conventional drones.
Of more than 100 Russian drones deployed on Tuesday, he said, 29 were jet-powered and 18 of those had been downed.
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