The phenomenon of the transformation of proteins into amyloid-fibrils is of interest, firstly, because it is closely connected to the so-called conformational diseases, many of which are hitherto incurable, and secondly, because it remains to be explained in physical terms (energetically and structurally). The process leads to fibrous aggregates in the form of extracellular amyloid plaques, neuro-fibrillary tangles and other intracytoplasmic or intranuclear inclusions. In this review, basic principles common to the field of amyloid fibril formation and conformational disease are underlined. Existing models for the mechanism need to be tested by experiment. The kinetic and energetic bases of the process are reviewed. The main controversial issue remains the coexistence of more than one protein conformation. The possible role of oligomeric intermediates, and of domain-swapping is also discussed. Mechanisms for cellular defence and novel therapies are considered.