The explosions reportedly struck an area between the University of Aleppo’s halls of residence and the architecture faculty on the first day of exams.
The Syrian Observatory for Human Rights put the death toll at 52, but Aleppo’s governor said 82 people had died.
State TV said “terrorists” had launched rockets at the campus, but activists blamed missiles fired by warplanes.
In almost two years of fighting in Syria, Aleppo has been the scene of intense conflict between government and opposition forces.
However, neither side has been able to force the other to retreat for good, says the BBC’s James Reynolds in neighboring Turkey.
Video footage of the aftermath of the explosions in Syria’s second city showed the facade of one of the university residences blown away.
Burned-out vehicles and bodies could be seen on the street outside, while tearful survivors were shown taking refuge in a nearby building.
The Syrian Observatory for Human Rights, a UK-based activist group, said at least 52 people had been killed, but added that the figure might rise dramatically because dozens were in a critical condition in hospital.
The governor of Aleppo, Mohammed Wahid Akkad, told the AFP news agency that so far 82 had died and more than 160 were wounded.
Mohammed Wahid Akkad blamed a “terrorist attack that targeted students on their first day of exams”.
A military source told AFP that a stray surface-to-air missile fired by rebels had hit the campus, which lies in a government-controlled area of the city. The nearest rebel-held area is more than a mile away.
However, no rebel group has said it was behind the blasts, and opposition activists said government had sent fighter jets to bomb the campus.
The Local Co-ordination Committees (LCC), an opposition activist network, posted a link to an online video which purported to show warplanes bombing the campus.
It showed students walking quickly away from the university after the first explosion. The camera then shakes to the sound of another explosion and people begin running.
“The warplanes of this criminal regime do not respect a mosque, a church or a university,” a student who gave his name as Abu Tayem told the Reuters news agency.
Rebels have previously carried out bombings against government targets in Aleppo. In October, at least 34 people were killed in a series of bombings in the city’s main square.
The United Nations says more than 60,000 people have been killed since the uprising against President Bashar al-Assad began in March 2011.
Meanwhile, Russia has rejected a request from more than 50 countries to have the UN Security Council refer the conflict in Syria to the International Criminal Court, calling the initiative “ill-timed and counterproductive”.
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