This thesis addresses nonlinear electrokinetic mechanisms for transporting fluid and particles in microfluidic devices for potential applications in biomedical chips, microelectronic cooling and microfuel cells. Nonlinear electrokinetics have many advantages, such as low voltage, low power, high velocity, and no significant gas formation in the electrolyte. However, they involve new and complex charging and flow mechanisms that are still not fully understood or explored.  Linear electrokinetic fingering that occurs when a fluid with a lower electrolyte concentration advances into one with a higher concentration is first analyzed. Unlike earlier miscible fingering theories, the linear stability analysis is carried out in the self-similar coordinates of the diffusing front. This new spectral theory is developed for small-amplitude gravity and viscous miscible fingering phenomena in general and applied to electrokinetic miscible fingering specifically. Transient electrokinetic fingering is shown to be insignificant in sub-millimeter micro-devices.  Nonlinear electroosmotic flow around an ion-exchange spherical granule is studied next. When an electric field is applied across a conducting and ion-selective porous granule in an electrolyte solution, a polarized surface layer with excess counter-ions is created. The flux-induced polarization produces a nonlinear slip velocity to produce micro-vortices around this sphere. This polarization layer is reduced by convection at high velocity. Two velocity scalings at low and high electric fields are derived and favorably compared with experimental results. A mixing device based on this mechanism is shown to produce mixing efficiency 10-100 times higher than molecular diffusion.  Finally, AC nonlinear electrokinetic flow on electrodes is studied. Two double layer charging mechanisms are responsible for the flow---one due to capacitive charging of ions from the bulk electrolyte and one due to Faradaic reactions at the electrode that consume or produce ions in the double layer. Faradaic charging is analyzed for specific reactions. From the theory, particular electrokinetic flows above the electrodes are selected for micro-pumps and bioparticle trapping by specifying the electrode geometry and the applied voltage and frequency.