March 29, 2017 0318 GMT
There was no immediate claim of responsibility for the blasts, the second wave of deadly attacks in the capital in less than a week after twin bombings killed 74 on Saturday, AFP wrote.
Wednesday's first attack saw a suicide bomber rush inside the building and blow himself up when police tried to prevent him from entering the courthouse in the center of Damascus, state media reported.
A police source said that 32 people were killed and 100 wounded.
The second blast hit a restaurant in the western Rabweh neighborhood, wounding 25 people, the source said.
"We were terrified because the sound of the explosion was enormous," a lawyer who was in the courthouse during the first attack said.
"We took refuge in the library which is on a higher floor," the lawyer said, speaking on condition of anonymity. "It was a bloody scene."
State television broadcast scenes from the building showing blood smeared on the floor of the lobby but also splattered across its ceiling.
It interviewed a man receiving treatment with a bandage over his eye who said the attacker was wearing a military jacket.
Damascus was already reeling from Saturday's bombings, which mainly killed Iraqi pilgrims in the city to visit Shia shrines.
That attack was claimed by former Al-Qaeda affiliate Fateh al-Sham Front, part of a terrorist alliance that controls large parts of the northwestern Idlib Province.
Terrorists suffered a series of reversals during the sixth year of the war, including being forced from their onetime stronghold of east Aleppo in December.
The loss was an especially difficult blow to terrorists who had imagined marching on Damascus in the early days of the war.
The conflict began in 2011, killing more than 320,000 people, with over the half the country's population displaced either within Syria or becoming refugees.
The war has also ravaged the country's infrastructure and set the economy back decades.