Theories of electro-osmotic slip, of either the equilibrium (first) or nonequilibrium (second) kind at a permselective conductive surface (ion exchange electrodialysis membrane, electrode) are reviewed. A slip condition for electro-osmosis of the second kind, relevant for a developed concentration polarization at an electrodialysis membrane, is derived through a boundary layer analysis of the appropriate convective electrodiffusion problem. Linear hydrodynamic stability of the quiescent concentration polarization in a diffusion layer at a cation exchange electrodialysis membrane is studied. It is shown that electro-osmotic slip of the second kind, as opposed to that of the first kind, yields instability for realistic conditions. Numerical calculations for the resulting nonlinear convection show that the latter provides an efficient mixing mechanism for the diffusion layer, capable of accounting for the overlimiting conductance in concentration polarization at a cation exchange membrane.