With the growing demand for radar target detection, Sparse Recovery (SR) technology based on the Compressive Sensing (CS) model has been widely used in radar signal processing. This paper first outlines the fundamental theory of SR and then introduces the sparse characteristics in radar signal processing from the perspectives of scene sparsity and observation sparsity. Subsequently, it explores these sparse properties to provide an overview of CS applications in radar signal processing, including spatial domain processing, pulse compression, coherent processing, radar imaging, and target detection. Finally, the paper summarizes the applications of CS in radar signal processing.