An overview is given of the physics issues relevant to the plasma wakefield accelerator, the plasma beat-wave accelerator, the laser wakefield accelerator, including the self-modulated regime, and wakefield accelerators driven by multiple electron or laser pulses. Basic properties of linear and nonlinear plasma waves are discussed, as well as the trapping and acceleration of electrons in the plasma wave. Formulas are presented for the accelerating field and the energy gain in the various accelerator configurations. The propagation of the drive electron or laser beams is discussed, including limitations imposed by key instabilities and methods for optically guiding laser pulses. Recent experimental results are summarized.

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[1] The Naval Research Laboratory (NRL) has developed a new three-dimensional code to study equatorial spread F (ESF). The code is based on the comprehensive NRL 3D ionosphere model SAMI3 and includes a potential equation to self-consistently solve for the electric field. The model assumes equipotential field lines so a 2D electrodynamic problem is considered. In this study a narrow wedge of the post-sunset ionosphere is simulated. It is found that (1) bubbles can rise to $1600 km, (2) extremely steep ion density gradients can develop in both longitude and latitude, (3) upward plasma velocities approach 1 km/s, and (4) the growth time of the instability is '15 min. These results are shown to be consistent with radar and satellite observations.