The partial unfolding or alternative folding of a class of polypeptides is at the origin of fascinating events in living cells. In their non-native conformation, these constitutive polypeptides called prions are at the origin of a protein-based structural heredity. These polypeptides are closely associated to a class of fatal neurodegenerative illnesses in mammals and to the emergence and propagation of phenotypic traits in baker's yeasts. The structural transition from the correctly folded, native form of a prion protein to a persistent misfolded form that ultimately may cause cell death or the transmission of phenotypic traits is not yet fully understood. The mechanistic models accounting for this structure-based mode of inheritance and the extent of partial unfolding of prions or their alternative folding and the subsequent aggregation process are developed and discussed. Finally, the potential regulation of prion propagation by molecular chaperones is presented.