Amyloids are filamentous protein aggregates that are associated with a number of incurable diseases, termed amyloidoses. Amyloids can also manifest as infectious or heritable particles, known as prions. While just one prion is known in humans and animals, more than ten prion amyloids have been discovered in fungi. The propagation of fungal prion amyloids requires the chaperone Hsp104, though in excess it can eliminate some prions. Even though Hsp104 acts to disassemble prion fibrils, at normal levels it fragments them into multiple smaller pieces, which ensures prion propagation and accelerates prion conversion. Animals lack Hsp104, but disaggregation is performed by the same complement of chaperones that assist Hsp104 in yeast—Hsp40, Hsp70, and Hsp110. Exogenous Hsp104 can efficiently cooperate with these chaperones in animals and promotes disaggregation, especially of large amyloid aggregates, which indicates its potential as a treatment for amyloid diseases. However, despite the significant effects, Hsp104 and its potentiated variants may be insufficient to fully dissolve amyloid. In this review, we consider chaperone mechanisms acting to disassemble heritable protein aggregates in yeast and animals, and their potential use in the therapy of human amyloid diseases.