The pits were found yesterday on a hill in a community outside Iguala, the town where the students were last seen and where witnesses say municipal police officers whisked several of them away.
Inaky Blanco, chief prosecutor for the violence-plagued state of Guerrero, declined to say how many bodies were buried in the pits outside Iguala, 200 kilometres (125 miles) south of Mexico City.
"We still can't talk about an exact number of bodies. We are still working at the site," Blanco told a news conference in the state capital, Chilpancingo.
Juan Lopez Villanueva, an official from the National Human Rights Commission, said that six pits were found up a steep hill probably inaccessible by car.
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Four forensic services vans left for the morgue late Thursday carrying bodies in silver bags. Authorities are conducting DNA analysis to identify the victims.
The graves were found after some of the 30 suspects detained in the case told authorities about their location, Blanco said. The detainees include 22 police officers and gang members.
The students from a teacher training college disappeared last weekend after Iguala police officers shot at buses that the group had seized to return home after holding fundraising activities on September 26. Three students were killed.
Another three people died when police and suspected gang members shot at another bus carrying football players on the outskirts of town.
A survivor said in an interview that the officers took away 30 to 40 students in patrol cars.
Authorities have issue an arrest warrant for Iguala's mayor, who has fled.