State television blamed "armed terrorist groups" for what were the first attacks in Aleppo since the outbreak of an uprising against the regime of President Bashar al-Assad almost a year ago.
The television said a "suicide bomber in a car packed with explosives" carried out one of the attacks on a police station, flattening a nearby food distribution centre.
The second bombing targeted an intelligence base.
The television showed mangled bodies in pools of blood in the street outside rows of shattered buildings. Emergency workers held up body parts, including hands, feet and a torso.
Soldiers were among the casualties, it said.
But the General Commission of the Syrian Revolution, an opposition group, said the attacks were "staged by the regime," in a statement emailed to AFP in Beirut.
The television footage was aired live from the site of the blasts, with bulldozers quickly deployed to clear away shattered concrete strewn across a broad avenue, revealing deep craters in the ground.
"A terrorist blew himself up with his car 100 metres (yards) from the entrance of the police station," it reported, adding that emergency services were scouring the rubble for bodies.
Several people who were interviewed denounced Turkey and Qatar for not standing by the regime in Syria as it seeks to put down a revolt that broke out last March.
"Is this the freedom they want?" asked one angry-looking man who was holding up a severed arm. He was referring to those leading the protest movement against Syria's government.
Rami Abdel Rahman, head of the Britain-based Syrian Observatory for Human Rights, told AFP there were two blasts in the northern city, after earlier reporting a total of three.
Aleppo has been largely spared the unrest that has rocked Syria for nearly a year, leaving more than 6,000 people dead according to rights groups.
Friday's blasts were the biggest since two suicide bombings in Damascus, one of which killed 44 people in December and the second 26 people in January.
The Assad regime also blamed those attacks on "armed terrorist gangs" but the opposition has blamed the government.
The bombings came ahead of planned protests on Friday by the opposition to denounce Russia's staunch support for the Assad regime.
Russia said on Friday that Syria's opposition bore full responsibility for the ongoing violence while accusing the West of pushing the regime's opponents into armed conflict.
Deputy Foreign Minister Sergei Ryabkov told ITAR-Tass news agency the opposition's refusal to enter direct talks with the Syrian government meant it "bears full responsibility for improving the situation."
He accused the West of being "accomplices in the process of inflaming the crisis."