The ordered assembly of tau protein into filaments characterizes several neurodegenerative diseases, which are called tauopathies. It was previously reported that, by cryo-electron microscopy, the structures of tau filaments from Alzheimer’s disease 1, 2, Pick’s disease 3, chronic traumatic encephalopathy 4 and corticobasal degeneration 5 are distinct. Here we show that the structures of tau filaments from progressive supranuclear palsy (PSP) define a new three-layered fold. Moreover, the structures of tau filaments from globular glial tauopathy are similar to those from PSP. The tau filament fold of argyrophilic grain disease (AGD) differs, instead resembling the four-layered fold of corticobasal degeneration. The AGD fold is also observed in ageing-related tau astrogliopathy. Tau protofilament structures from inherited cases of mutations at positions+ 3 or+ 16 in intron 10 of MAPT (the microtubule-associated protein tau gene) are also identical to those from AGD, suggesting that relative overproduction of four-repeat tau can give rise to the AGD fold. Finally, the structures of tau filaments from cases of familial British dementia and familial Danish dementia are the same as those from cases of Alzheimer’s disease and primary age-related tauopathy. These findings suggest a hierarchical classification of tauopathies on the basis of their filament folds, which complements clinical diagnosis and neuropathology and also allows the identification of new entities—as we show for a case diagnosed as PSP, but with filament structures that are intermediate between those of globular glial tauopathy and PSP.