Active accumulation of the data on new amyloids continuing nowadays dissolves boundaries of the term "amyloid". Currently, it is most often used to designate aggregates with cross-β structure. At the same time, amyloids also exhibit a number of other unusual properties, such as: detergent and protease resistance, interaction with specific dyes, and ability to induce transition of some proteins from a soluble form to an aggregated one. The same features have been also demonstrated for the aggregates lacking cross-β structure, which are commonly called "amyloid-like" and combined into one group, although they are very diverse. We have collected and systematized information on the properties of more than two hundred known amyloids and amyloid-like proteins with emphasis on conflicting examples. In particular, a number of proteins in membraneless organelles form aggregates with cross-β structure that are morphologically indistinguishable from the other amyloids, but they can be dissolved in the presence of detergents, which is not typical for amyloids. Such paradoxes signify the need to clarify the existing definition of the term amyloid. On the other hand, the demonstrated structural diversity of the amyloid-like aggregates shows the necessity of their classification.