This paper gives a brief review on the Sparse-Recovery (SR)-based Space-Time Adaptive Processing (STAP) technique. First, the motivation for introducing sparse recovery into STAP is presented. Next, the potential advantages and mathematical explanation of the sparse-recovery-based STAP are discussed. A major part of this paper presents the state-of-art research results in spatio-temporal spectrum-sparsity-based STAP, including the basic frame, off-grid problem, multiple measurement vector problem, and direct domain problem. The sparse-recovery-based STAP on conformal array problem is also introduced. Finally, a summary of sparse-recovery-based STAP is provided, and the problems that need to be solved and some potential research areas are discussed.