The black bin bags were discovered yesterday in a suburb of the commercial capital Dar es Salaam.
"We have arrested eight suspects in connection with the 85 bags found in Kinondoni area of Dar es Salaam," city police chief Suleiman Kova told AFP.
Those arrested were reported to work at a private medical teaching college.
"We are detaining them for interrogation over the alleged dumping of the human body parts," Kova added.
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Kova said limbs, fingers and skulls were all found in the bags, but that the flesh was not fresh.
"I can't say how many human bodies were there, but we have taken them to Muhimbili National Hospital for further testing," he said. "They were dry and produced no smell."
Police said they had no information as to why the body parts had been dumped, and would not comment as to whether they were connected to witchcraft ceremonies.
Attackers in Tanzania have killed a string of albinos in recent years. In Tanzania, albinos are killed and dismembered due to a widespread belief that charms made from their body parts bring good fortune and prosperity.