  We revisit the classical problem of diffusion-limited ion transport to a
membrane (or electrode) by considering the effects of charged side walls. Using
simple mathematical models and numerical simulations, we identify three basic
mechanisms for over-limiting current in a microchannel: (i) {\it surface
conduction} carried by excess counterions, which dominates for very thin
channels, (ii) convection by {\it electro-osmotic flow} on the side walls,
which dominates for thicker channels and transitions to (iii) {\it
electro-osmotic instability} on the membrane end in very thick channels. These
intriguing electrokinetic phenomena may find applications in biological
separations, water desalination, and electrochemical energy storage.
