The prion hypothesis proposes that proteins can act as infectious agents. Originally formulated to explain transmissible spongiform encephalopathies (TSEs), the prion hypothesis has been extended with the finding that several non-Mendelian traits in fungi are due to heritable changes in protein conformation, which may in some cases be beneficial. Although much remains to be learned about the specific role of cellular cofactors, mechanistic parallels between the mammalian and yeast prion phenomena point to universal features of conformation-based infection and inheritance involving propagation of ordered beta-sheet-rich protein aggregates commonly referred to as amyloid. Here we focus on two such features and discuss recent efforts to explain them in terms of the physical properties of amyloid-like aggregates. The first is prion strains, wherein chemically identical infectious particles cause distinct phenotypes. The second is barriers that often prohibit prion transmission between different species. There is increasing evidence suggesting that both of these can be manifestations of the same phenomenon: the ability of a protein to misfold into multiple self-propagating conformations. Even single mutations can change the spectrum of favored misfolded conformations. In turn, changes in amyloid conformation can shift the specificity of propagation and alter strain phenotypes. This model helps explain many common and otherwise puzzling features of prion inheritance as well as aspects of noninfectious diseases involving toxic misfolded proteins.