The primary progressive aphasias (PPA) are paradigmatic disorders of language network breakdown associated with focal degeneration of the left cerebral hemisphere. Here we addressed brain correlates of PPA in a detailed neuroanatomical analysis of the third canonical syndrome of PPA, logopenic/phonological aphasia (LPA), in relation to the more widely studied clinico-anatomical syndromes of semantic dementia (SD) and progressive nonfluent aphasia (PNFA). 32 PPA patients (9 SD, 14 PNFA, 9 LPA) and 18 cognitively normal controls had volumetric brain MRI with regional volumetry, cortical thickness, grey and white matter voxel-based morphometry analyses. Five of nine patients with LPA had cerebrospinal fluid biomarkers consistent with Alzheimer (AD) pathology (AD-PPA) and 2/9 patients had progranulin (GRN) mutations (GRN-PPA). The LPA group had tissue loss in a widespread left hemisphere network. Compared with PNFA and SD, the LPA group had more extensive involvement of grey matter in posterior temporal and parietal cortices and long association white matter tracts. Overlapping but distinct networks were involved in the AD-PPA and GRN-PPA subgroups, with more anterior temporal lobe involvement in GRN-PPA. The importance of these findings is threefold: firstly, the clinico-anatomical entity of LPA has a profile of brain damage that is complementary to the network-based disorders of SD and PNFA; secondly, the core phonological processing deficit in LPA is likely to arise from temporo-parietal junction damage but disease spread occurs through the dorsal language network (and in GRN-PPA, also the ventral language network); and finally, GRN mutations provide a specific molecular substrate for language network dysfunction.