A broadcast described the blasts as “terrorist” attacks. Preliminary reports suggested vehicles packed with explosives had been detonated, it said.
It said intelligence and police buildings were hit and the cause was not known.
Details of the reports cannot be independently verified as access to Syria for journalists is restricted.
Dozens of people have been killed in bomb attacks in Damascus and the second city Aleppo in recent months, which the government also blamed on terrorists.
The opposition has accused the authorities of staging some of those incidents.
State TV showed pictures of charred bodies, burned-out vehicles and bloodstains on the streets.
It described one body as being that of a terrorist.
It said buildings housing the criminal police and aviation intelligence had been targeted.
Opposition sources also said security buildings had been hit.
Fresh anti-government protests were held on Friday in cities across Syria.
And there was a return of violence to the Damascus suburbs – the first significant fighting there since government forces imposed military control some weeks ago.
Clashes between rebel fighters and the army were reported in several other parts of the country.
President Bashar al-Assad insists his troops are fighting “armed gangs” seeking to destabilize Syria.
On Friday, UN and Arab League special envoy Kofi Annan renewed calls for an end to fighting and for unimpeded humanitarian aid for Syria.
Speaking to UN Security Council members, Kofi Annan said he was sending a team to Damascus to discuss setting up a new international monitoring mission.
The international community remains divided on Syria, with Russia and China both blocking UN Security Council resolutions on Syria and aid groups from 27 countries urging them to condemn the government’s use of violence.
But the two permanent members have backed Kofi Annan’s peace mission.
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