Electrokinetically driven flow of electrolyte solutions through micro- and nanochannels is of interest in microelectromechanical systems (MEMS) and nanotechnology applications. In this work, fully developed and steady electroosmotic flow (EOF) of dilute sodium tetraborate and sodium chloride aqueous solutions in a rectangular channel where the channel hight h is comparable to its width W is examined. EOF is also studied under conditions of electric double layer (EDL) overlap, or /h O(1), where is the Debye thickness, for very dilute solutions. The initial experimental data and model results are in very good agreement for dilute sodium tetraborate solutions. The experimental work uses the new nano-particle image velocimetry (nPIV) technique. Evanescent waves from the total internal reflection of light with a wavelength of 488 nm at a refractive index interface is used to illuminate 100 nm neutrally buoyant fluorescent particles in the near-wall region of the flow. The images of these tracer particles over time are processed to obtain the two components of the velocity field parallel to the wall in fully developed EOF of sodium tetraborate at concentrations up to 2 mM in fused quartz rectangular channels with height h up to 10 microns. The spatial resolution of these velocity field data along the dimension normal to the wall is about 100 nm, and the data are obtained within a distance of approximately 100 nm of the wall based upon the 1/e intensity point, or penetration depth. A set of equations modeling EOF in a long channel are solved where h/L >> 1, and L is the lengthscale along the flow direction. Unlike most previous models, this work does not use the Debye-Huckel approximation, nor does it assume symmetric boundary conditions. For the case where /h >> 1, analytical solutions for the velocity, potential and mole fractions are obtained using an asymptotic perturbation approach.