The low permeability of clays, shales, and other argillaceous lithologies makes them key controls of transport and deformation processes in the crust but is known for being challenging to characterize. As muds are modified by compaction and diagenesis to low-porosity shales, permeability can decrease by six or more orders of magnitude, but at large scales it is often dramatically and unpredictably increased by fractures, faults, and other features. Testing and inverse modeling show that petrophysical properties and the geological environment are dominant controls of clay and shale matrix permeability and its scale dependence. Active sedimentation and tectonism on continental margins cause large-scale permeability to vary with time, but in stable continent interiors it is unclear how regional permeability of argillaceous formations changes over time or, in most cases, what controls it. Although rarely considered, it is also unknown whether Darcian permeability adequately describes flow in clay-rich materials. ▪ Critical for problems in energy, water supply, waste isolation, and geologic hazards, clay and shale permeability remains problematic. ▪ Test data and inverse model analyses are beginning to reveal where and how permeability of clay and shale changes with scale. ▪ In clays and shales, causes of permeability scale effects, their time dependence, and even flow behavior continue to raise questions.