Electrical control of liquid droplet motion and wettability has wide-ranging applications in the field of MEMS, lab-on-a-chip devices and surface engineering, in view of the resulting enhanced flow control opportunities, low power consumption and the absence of mechanical moving parts. This article summarizes recent progress towards understanding of the fundamentals underlying electrical actuation of droplets on smooth and superhydrophobic surfaces. Electrical actuation of liquid droplets with widely differing electrical properties on smooth surfaces is first discussed. Electromechanical considerations are employed to study the actuation force on a generic liquid droplet across the entire spectrum of electrical actuation regimes. The challenges in understanding the fluid flow and dissipation mechanisms associated with a discrete moving droplet are discussed. The role of electrical voltages, interfacial energies and surface morphology in determining droplet states (nonwetting Cassie state and wetting Wenzel state) and triggering state transitions on superhydrophobic surfaces is then mapped out. Critical phenomena associated with droplet transitions on superhydrophobic surfaces (energy barrier for the Cassie-Wenzel transition, lack of spontaneous reversibility of the Cassie-Wenzel transition, robustness of the Cassie state, and the role of the roughness elements) are analyzed. The article also highlights key avenues for future research in the fields of electrical actuation-based microfluidics and superhydrophobic surfaces.