The Syrian Observatory for Human Rights said at least three people died when several rounds hit Jaramana, a mainly Christian suburb that is considered pro-regime.
And an eight-year-old girl was killed and 11 others were wounded in rebel mortar fire on the Abu Rummaneh district of the capital, state news agency SANA reported.
The Observatory said she was in a car when hit.
Rebel forces have increasingly targeted central districts of Damascus with mortar rounds, firing from their rear bases on the outskirts of the capital.
Elsewhere, the Syrian Observatory, a Britain-based NGO, said government forces dropped "barrel bombs" on the strategic town of Sfeireh in the northern province of Aleppo.
The town is largely controlled by jihadist fighters and is near a military site that is believed to hold some of the regime's arsenal of chemical weapons.
"The regime wants to retake Sfeireh because it wants to take the inspectors to the site," Observatory director Rami Abdel Rahman told AFP.
Two teams of chemical weapons inspectors are in Damascus verifying and destroying Syria's chemical weapons arsenal under the terms of a UN resolution.
Underlining the dangers also faced by foreign journalists covering the conflict, a Russian photographer has been abducted by a group of rebels who accuse him of being a spy, the Russian foreign ministry said.
Ministry spokesman Alexander Lukashevich said photographer Konstantin Zhuravlev was seized in Aleppo and was being held by the Islamist group Liwa al-Tawhid.
The 32-year-old had hitchhiked from Siberia and entered Syria via Turkey. He was thought to be travelling through Syria en route to the Sahara desert.
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