The plasma synthetic jet actuator (PSJA), also named as sparkjet actuator, is a special type of zero-net mass flux actuator, driven thermodynamically by pulsed arc/spark discharge. Compared to widely investigated mechanical synthetic jet actuators driven by vibrating diaphragms or oscillating pistons, PSJAs exhibit the unique capability of producing high-velocity (>300 m/s) pulsed jets at high frequency (>5 kHz), thus tailored for high-Reynolds-number high-speed flow control in aerospace engineering. This paper reviews the development of PSJA in the last 15 years, covering the major achievements in the actuator working physics (i.e., characterization in quiescent air) as well as flow control applications (i.e., interaction with external crossflow). Based on the extensive non-dimensional laws obtained in characterization studies, it becomes feasible to design an actuator under several performance constraints, based on first-principles. The peak jet velocity produced by this type of actuator scales approximately with the cubic root of the non-dimensional energy deposition, and the scaling factor is determined by the electro-mechanical efficiency of the actuator (O(0.1%โ1%)). To boost the electro-mechanical efficiency, the energy losses in the gas heating phase and thermodynamic cycle process should be minimized by careful design of the discharge circuitry as well as the actuator geometry. Moreover, the limit working frequency of the actuator is set by the Helmholtz natural resonance frequency of the actuator cavity, which can be tuned by the cavity volume, exit orifice area and exit nozzle length. In contrast to the fruitful characterization studies, the application studies of PSJAs have progressed relatively slower, not only due to the inherent difficulties of performing advanced numerical simulations/measurements in high-Reynolds-number high-speed flow, but also related to the complexity of designing a reliable discharge circuit that can feed multiple actuators at high repetition rate. Notwithstanding these limitations, results from existing investigations are already sufficient to demonstrate the authority of plasma synthetic jets in shock wave boundary layer interaction control, jet noise mitigation and airfoil trailing-edge flow separation.