The fundamentals of observing moving targets with synthetic aperture radar (SAR) are reviewed. Phase equations for moving point targets are derived. Included are equations for the spreading of target energy into multiple azimuth (i.e., cross range, Doppler) resolution cells due to uncompensated target acceleration (range direction) and velocity (azimuth direction). The phenomena of target shift or translation in the azimuth direction of the image due to range velocity is also derived. Examples of moving target phenomenology which include actual measured phase responses are given. Space-time adaptive processing (STAP), applied to multiple channel SAR (element space) for the purpose of detecting and locating moving targets is reviewed. The associated STAP signal-to-interference ratio (SINR) to be maximized is introduced. It is observed that clutter (ground reflectivity) suppression associated with STAP is a result of applying the inverse of the computed “clutter plus noise covariance matrix” to a stack of SAR images. Target location estimation as a byproduct of SINR maximization is also discussed. The talk is intended for a general audience.