Infectious proteins (prions) became an important medical issue when they were identified as agents of the transmissible spongiform encephalopathies. More recently, prions have been found in fungi and their investigation has been facilitated by greater experimental tractability. In each case, the normal form of the prion protein may be converted into the infectious form (the prion itself) in an autocatalytic process; conversion may either occur spontaneously or by transmission from an already infected cell. Four fungal prion proteins have been studied in some depth-Ure2p, Sup35p, and Rnq1p of Saccharomyces cerevisiae and HET-s of Podospora anserina. Each has a "prion domain" that governs infectivity and a "functional domain" that contributes the protein's activity in a wild-type cell, if it has one. This activity is repressed in prion-infected cells for loss-of-activity prions, [URE3] (the prion of Ure2p) and [PSI] (the prion of Sup35p). For gain-of-activity prions, [PIN] (the prion of Rnq1p) and [Het-s] (the prion of HET-s), the prion domain is also involved in generating a new activity in infected cells. In prion conversion, prion domains polymerize into an amyloid filament, switching from a "natively unfolded" conformation into an amyloid conformation (stable, protease-resistant, rich in cross-beta structure). For Ure2p and probably also Sup35p, the functional domain retains its globular fold but is inactivated by a steric mechanism. We review the evidence on which this scenario is based with emphasis on filament structure, summarizing current experimental constraints and appraising proposed models. We conclude that the parallel superpleated beta-structure and a specific beta-helical formulation are valid candidates while other proposals are excluded. In both the Ure2p and Sup35p systems, prion domain amyloid filaments exhibit polymorphic variation. However, once a certain structure is nucleated, it is maintained throughout that filament. Electron microscopy of several Ure2p-related constructs indicates that the basis for polymorphism lies mainly if not entirely in the prion domain. Filament polymorphism appears to underlie the phenomenon of prion "variants" which differ in the severity of their phenotype, that is, for Ure2p and Sup35p, the stringency with which their activity is switched off. We discuss a possible structural basis for this phenomenon.