The main focus of the review is on dynamics and kinetics of near-surface discharge plasmas, such as surface dielectric barrier discharges sustained by AC and repetitively pulsed waveforms, pulsed DC discharges, and quasi-DC discharges, generated in quiescent air and in the airflow. A number of technical issues related to plasma flow control applications are discussed in detail, including discharge development via surface ionization waves, charge transport and accumulation on dielectric surface, discharge contraction, different types of flow perturbations generated by surface discharges, and effect of high-speed flow on discharge dynamics. In the first part of the manuscript, plasma morphology and results of electrical and optical emission spectroscopy measurements are discussed. Particular attention is paid to dynamics of surface charge accumulation and dissipation, both in diffuse discharges and during development of ionization instabilities resulting in discharge contraction. Contraction leads to significant increase of both the surface area of charge accumulation and the energy coupled to the plasma. The use of alternating polarity pulse waveforms accelerates contraction of surface dielectric barrier discharges and formation of filamentary plasmas. The second part discusses the interaction of discharge plasmas with quiescent air and the external airflow. Four major types of flow perturbations have been identified:(1) low-speed near-surface jets generated by electrohydrodynamic interaction (ion wind);(2) spanwise and streamwise vortices formed by both electrohydrodynamic and thermal effects;(3) weak shock waves produced by rapid heating in pulsed discharges on sub-microsecond time scale; and (4) near-surface localized stochastic perturbations, on sub-millisecond time, detected only recently. The mechanism of plasma-flow interaction remains not fully understood, especially in filamentary surface dielectric barrier discharges. Localized quasi-DC surface discharges sustained in a high-speed flow are discussed in the third part of the review. Although dynamics of this type of the discharge is highly transient, due to its strong interaction with the flow, the resultant flow structure is stationary, including the oblique shock and the flow separation region downstream of the discharge. The oblique shock is attached to a time-averaged, wedge-shaped, near-wall plasma layer, with the shock angle controlled by the discharge power, which makes possible changing the flow structure and parameters in a controlled way. Finally, unresolved and open-ended issues are discussed in the summary.