In northern Syria, 14 children were among 25 people killed in an air strike on Idlib city, a monitor said.
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 Saturday that killed 74 people.
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 centre of Damascus, state media reported.
A police source told AFP that 32 people were killed and 100 wounded.
"I heard a commotion and looked to my left and I saw a man in a military vest," a man with a bandage over his eye told state television after the attack.
"He had his hands up and screamed 'God is greatest' and then the blast happened," he added.
State television broadcast images from inside the courthouse, showing blood splattered across the ceiling and smeared across the marble floor of the lobby, with a portrait of President Bashar al-Assad still intact and hanging above.
Streets deserted
The second blast hit a restaurant in the city's western Rabweh district less than two hours later, and injured 25 people, the police source said.
State media said the bomber had ducked into the restaurant after being chased by security services.
In the wake of the attacks, AFP correspondents in the city said the streets were deserted, with some roads blocked off by security services.
Elsewhere, 25 people including 14 children were killed in an air strike in Idlib city in northwestern Syria, according to the Syrian Observatory for Human Rights.
More than 320,000 people have been killed in Syria since the conflict began in March 2011, with Assad's government now holding the upper hand in the bloody war and rebel forces increasingly divided and dispirited.
In recent months, the opposition has suffered a series of reversals, including being forced from their one-time stronghold of east Aleppo in December.
And peace talks have made little progress towards resolving the conflict, with rebels declining to even attend a latest round of negotiations in Kazakhstan which wrapped up Wednesday.
The conflict began in 2011 with peaceful demonstrations inspired by similar movements during the so-called "Arab Spring", calling on Assad to implement reforms.
They were put down violently, prompting demonstrators to pick up weapons as the uprising spiralled into a complex, brutal civil war.
IS in retreat
Rebel forces initially captured large parts of the country and several key cities, and won support from international backers including Washington, Turkey and the Gulf states.
But the Islamic State jihadist group emerged from the chaos to seize control of significant territory in Syria and neighbouring Iraq.
A US-backed Arab-Kurdish alliance, Turkish troops and allied rebels, and government troops aided by Russia have all squeezed northern Syrian territory from the jihadists.
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Six years and counting: Syria's civil war
In September 2015, government ally Russia began a military intervention in support of Assad, helping his forces regain much of the ground they had lost.
Washington said Wednesday that American and Russian troops were now both present in the northern town of Manbij, which fell from IS hands last year.
"They can see each other. They are not talking to each other, and they are not hanging out together," military spokesman Colonel John Dorrian told reporters.
'Savage horror'
On Wednesday, Aleppo residents said water was gradually being restored after months of cuts after IS was expelled from a local pumping station by a government offensive.
State television said one of its cameramen, Ali Suleiman, was killed when its team in east Aleppo province was "targeted by Daesh terrorists," using an Arabic acronym for IS, without giving further details.
The war's brutality has provoked international outcry, with the UN's rights chief Zeid Ra'ad Al Hussein this week describing Syria as "a place of savage horror and absolute injustice".
UN Secretary General Antonio Guterres on Wednesday urged adherence to a ceasefire brokered by Russia and Turkey in December, and progress towards peace, which he called "a moral and political imperative both for the Syrian people and for the world".
Peace is something that ordinary Syrians, including many who supported the uprising, are desperate for.
"When we began to demonstrate, I never thought it would come to this. We thought it would end in two, three months, a year at most," Abdallah al-Hussein, a 32-year-old footballer from the town of Saraqeb in Idlib province, told AFP.
"Whether this war is ended with weapons or peacefully doesn't matter. People want to live in peace."
US plans for Syria include another 1000 troops: US official
Up to 1000 additional US troops could deploy to northern Syria under provisional plans drawn up by the Pentagon, a US defense official told AFP on Wednesday.
The plans, which still need to be approved by President Donald Trump and Defense Secretary Jim Mattis, would mark a significant uptick in US boots on the ground in Syria as part of the fight against the Islamic State group.
Currently, the troop level is capped at 500 in Syria, but that number has become increasingly meaningless as commanders flow extra "temporary" forces in as needed - such as last week's deployment of a Marine artillery battery near Raqa.
The actual number of American troops in the war-torn country is likely now between 800 and 900, and a US defense official said the new plans would allow for up to 1000 more.
"That's one of the proposals that's on the table for discussion," the official said, speaking on condition of anonymity.
The possible deployment was first reported by the Washington Post, which said the extra forces would come from the 24th Marine Expeditionary Unit and the US Army's 82nd Airborne Division.
The official said the troops would not be directly in combat, but rather in support roles for any additional capabilities the military requires in northern Syria, where a US-led coalition is training and backing a local Kurdish-Arab alliance to fight the Islamic State group.
Such missions could include additional artillery batteries and the use of rocket launchers known as HIMARS that might be used to provide round-the-clock bombardment in the battle to recapture Raqa from IS.
Former president Barack Obama was loath to deploy combat troops into Syria and Iraq to fight IS, arguing the battle could only be meaningfully won by local forces.
Trump has said he wants to quicken the defeat of IS and told the Pentagon to come up with a range of plans that could accomplish that goal.
