Compressed sensing is a new paradigm in signal processing that trades sampling frequency for computing power and allows accurate reconstruction of signals sampled at rates many times less than the conventional Nyquist frequency. Today, modern radar systems operate with high bandwidths and high resolution. Compared to complex radar system and mass data, often only a small amount of target parameters is the final output. Compressed sensing is one of good means to effectively reduce data size. This paper reviews the latest developments of compressed sensing in radar target detection and recognition and introduces the key technical problems of design of measurement matrix and reconstruction algorithm for sparse signal. Several possible applications are considered: PD radar, through wall radar, MIMO radar, radar target parameter estimation, radar imaging and radar target detection and recognition system. Then this paper also discusses the existing difficult problems in the study and looks into the future research directions on compressive sensing applied to radar.