Below the seismogenic zone, plate boundaries are defined by lithospheric ductile shear zones. These localize strain, and are the main reason that convective motions in the Earth's interior are expressed at the surface as plate tectonics. We argue here that lithospheric shear zones operate at approximately constant stress, equal to the yield strength of the surrounding rocks. This places constraints on the bulk strength of the lithosphere, and allows us to calculate the cumulative width of the shear zones as a function of depth. The concept applies most clearly to strike-slip shear zones, but is also applicable to thrust-and normal-sense shear zones, although these evolve with time due to temperature changes associated with burial or exhumation. If shear zones operate at constant stress, this affects their microstructural evolution, and hence their rheology. Decreases in grain size due to dynamic recrystallization may cause a switch in the dominant deformation mechanism to grain-size-sensitive creep, leading to weakening and strain localization. A common argument, based on a constant strain rate approach, has been that such transitions will be inhibited by grain growth. In a constant stress shear zone, however, dynamic recrystallization continues to maintain the low grain size even after the switch has occurred. Grain growth is inhibited under these conditions, and hence the switch is permanent as long as the boundary conditions remain the same. Grain-size reduction in shear zones is therefore a critical factor in maintaining plate tectonic processes.