Amyloid fibrils are elongated protein aggregates well known for their association with many human diseases. However, similar structures have also been found in other organisms and amyloid fibrils can also be formed in vitro by other proteins usually under non-physiological conditions. In all cases, these fibrils assemble in a nucleated polymerization reaction with a pronounced lag phase that can be eliminated by supplying pre-formed fibrils as seeds. Once formed, the fibrils are usually very stable, except for their tendency to break into smaller pieces forming more growing ends in the process. These properties give amyloid fibers a self-replicating character dependent only on a source of soluble protein. For some systems and under certain circumstances this can lead to infectious protein structures, so-called prions, that can be passed from one organism to another as in the transmissible spongiform encephalopathies and in fungal prion systems. Structural details about these processes have emerged only recently, mostly on account of the inability of traditional highresolution methods to deal with insoluble, filamentous specimens. In consequence, current models for amyloid fibrils are based on fewer constraints than common atomic-resolution structures. This review gives an overview of the constraints used for the development of amyloid models and the methods used to derive them. The principally possible structures will be introduced by discussing current models of amyloid fibrils from Alzheimers β-peptide, amylin and several fungal systems. The infectivity of some amyloids under specific conditions might not be due to a principal structural difference between infectious and non-infectious amyloids, but could result from an interplay of the rates for filament nucleation, growth, fragmentation, and clearance.