Amyloids are highly ordered aggregates of protein fibrils exhibiting cross-β structure formed by intermolecular hydrogen bonds. Pathological amyloid deposition is associated with the development of several socially significant incurable human diseases. Of particular interest are infectious amyloids, or prions, that cause several lethal neurodegenerative diseases in humans and can be transmitted from one organism to another. Because of almost complete absence of criteria for infectious and non-infectious amyloids, there is a lack of consensus, especially, in the definition of similarities and differences between prions and non-infectious amyloids. In this review, we formulated contemporary molecular-biological criteria for identification of prions and non-infectious amyloids and focused on explaining the differences between these two types of molecules.