Keywords: electrokinetics, nanoscience, limiting current, Donnan potential, ion selectivity, Warburg Impedance Abstract: With the advent of nanofabrication technologies, nano-channels with dimensions smaller than the Debye screening layer can now be fabricated to allow scrutiny of the various anomalous DC and AC I-V characteristics of ion-selective membranes at the single-pore level — such knowledge is essential for rapid DNA sequencing, single-molecule sensing/identification and plasmonic imaging in nanoscience. Combining theoretical analyses of the underlying ion/solvent fluxes and confocal imaging of velocity and ion concentration fields, we explore the fundamental mechanisms behind non-ideal selectivity, Donnan potential, asymmetric depletion/enrichment layer formation, limiting and overlimiting-current, diode-like rectification, Warburg impedance response, inter-channel communication etc. Curiously, hydrodynamic effects at the depletion end of the channel is found to control many of the non-Ohmic behavior at higher voltages. Interfacial vortices created by an osmotic pressure driven instability (first predicted by I. Rubinstein) and induced charges at the corners of nanopores are found to specify the overlimiting current, the rectification factor and inter-pore communication. The intensity of these vortices and their influence on the ion-carried currents are found to be strongly dependent on the pore/reservoir geometries and can be described by limiting fundamental solutions of the Laplace and Stokes equations due to severe electric and flow field focusing into the nanochannel.