Electroosmotic flow of salt-free power-law fluids through planar slit and cylindrical micro and nanochannels with fluid slip is theoretically analyzed. Analytical solutions are obtained to investigate the effects of flow behavior index, channel size, applied electric field strength, Gouy-Chapman length (or surface charge density), and fluid slip length on the velocity distribution and volumetric flow rate. The results show that the electroosmotic flow velocity and thereby the flow rate for shear-thinning fluids are many times larger than those for Newtonian and shear-thickening fluids for the ranges of applied electric field strength and surface charge density usually encountered in practice. Such augmentation can be further amplified by increasing the surface charge density, applied electric field strength and fluid slip. Furthermore, the electroosmotic flow velocity profile of shear-thinning fluids becomes more plug-like as the ratio of channel half-width (or radius) to Gouy-Chapman length increases. However, such a profile for shear-thickening fluids always exhibits a parabolic-like flow pattern regardless of the ratios of channel half-width (or radius) to Gouy-Chapman length.