Museum of Paleontology: Located in the west wing of the Santi Marcellino e Festo Cloister, the museum houses more than 50,000 artifacts, the oldest dating back 600 million years. The first finds to be displayed were the Ittioliti deposits of Campania, fossils found by Oronzio Gabriele Costa (founder of Italian Paleontology).
Royal Museum of Mineralogy: Located inside the University itself, there's a Vesuvian collection, a collection of scientific instruments, and two hyaline quartz from Madagascar given as a present to King Charles of Bourbon.
Zoological Museum: One thousand skin specimens date back to the 18th century and several extinct animals are on display, including the Norfolk Island Pigeon, the Crescent Nail-Tailed Wallabi, and the Berber Lion. They also have a 30,000 specimens of insects and 2,000 specimens of parasitic worms.
Museum of Anthropology: Archeological finds from Troy are housed here, including polished stone axes and milling querns from about 3,000 B.C.
Biblioteca Universita: More difficult to get into, ask the lady at the top of the stairs if there are any exhibits to visit. They usually have some exhibitions that curators are more than willing to show you. (I was there when they were showing medieval maps.) The library itself contains about one million volumes, but the books themselves are hidden in the clavicles of the building, accessible only to librarians.
Museum of Paleontology Entrance
University entrance at Corso Umberto