In this work the electro-osmotic flow in a rectangular channel such that the channel height is comparable to its width is examined. Almost all previous work on the electro-osmotic flow in a channel has been for the case where the channel width is much greater than the channel height and the flow is essentially one-dimensional and depends only on channel height. We consider a mixture of water or another neutral solvent and a salt compound such as sodium chloride for which the ionic species are entirely dissociated. Results are produced for the case where the channel height is much greater than the electric double layer (EDL) (microchannel) and for the case where the channel height is of the order of the width of the EDL (nanochannel). Both symmetric and asymmetric velocity, potential and mole fraction distributions are considered, unlike previous work on this problem. In the symmetric case where all quantities are symmetric about the centreline, the velocity field and the potential are identical as in the parallel-plate one-dimensional case. In the asymmetric case corresponding to different wall potentials, the velocity and potential can be vastly different and reversed flow can occur. The results indicate that the Debye layer thickness is not a good measure of the actual width of the electric double layer. The binary results are shown to compare well with experiment and asymptotic solutions are also obtained for the case of a three-component mixture which may be applied to biomolecular transport.