According to the Syrian Observatory for Human Rights, rockets fired by regime forces rained down on Douma, one of the largest towns in the opposition-held Eastern Ghouta region.
Observatory head Rami Abdel Rahman said air strikes also hit Douma and the nearby town of Saqba, but it was unclear whether they were carried out by Syrian or Russian warplanes.
The air strikes and rockets wounded dozens, he said.
"One of the air strikes on Douma hit near a school, killing the school's principal," Abdel Rahman told AFP.
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A young girl was killed and three people were wounded in rebel mortar fire on Damascus, according to the national news agency SANA.
Last month, regime forces and rebel groups tried to reach a 15-day ceasefire in Eastern Ghouta, but the talks failed and violence has resumed.
A local activist group in Douma shared photographs on Facebook of crumbling buildings and bloodied children lying in a makeshift clinic.