The paper provides an overview of recent progress in the use of surface dielectric barrier discharges sustained by repetitive, high-voltage, nanosecond duration pulses for high-speed flow control. Experimental studies of diffuse and filamentary surface nanosecond pulse discharges in quiescent air demonstrate that they generate compression waves, due to rapid localized heating produced in the plasma. Compression waves produced by individual discharge filaments have higher amplitude and higher speed compared with waves produced in a diffuse discharge. Unlike surface dielectric barrier discharges sustained by AC voltage waveforms, nanosecond pulse discharges transfer little momentum to quiescent air, suggesting that localized heating and subsequent compression wave formation is the dominant flow control mechanism. Flow separation control using a nanosecond pulse surface discharge plasma actuator on an airfoil leading edge is studied up to M=0.26, Re=1.15·106 (free stream flow velocity 93 m/s), over a wide range of angles of attack. At pre-stall angles of attack, the actuator acts as an active boundary layer trip. At post-stall angles of attack, strong flow perturbations generated by the actuator excite shear layer instabilities and generate coherent spanwise vortices. These coherent structures entrain freestream momentum, thereby reattaching the separated flow to the suction surface of the airfoil. Feasibility of supersonic flow control by low-temperature nanosecond pulse plasma actuators is demonstrated in Mach 5 air flow over a cylinder model. Strong perturbations of a bow shock standing in front of the model are produced by compression waves generated in the plasma. Interaction of the compression waves and the bow shock causes its displacement in the upstream direction, increasing shock stand-off distance by up to 25%. The effect of compression waves generated by nanosecond discharge pulses on shock stand-off distance is demonstrated for single-pulse and quasi-continuous actuator operation. A self-similar kinetic model is developed to analyze energy coupling to the plasma in a surface ionization wave discharge produced by a nanosecond voltage pulse. The model predicts key discharge parameters such as ionization wave speed and propagation distance, electric field, electron density, plasma layer thickness, and pulse energy coupled to the plasma, demonstrating good agreement with available experimental data and two-dimensional kinetic modeling calculations. The model allows an analytic solution and lends itself to incorporating into existing compressible flow codes, for in-depth analysis of the nanosecond discharge plasma flow control mechanism.