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Array signal processing is an essential tool in broad radar applications. The coprime array has recently been proposed to overcome the bottleneck caused by the Nyquist spatial sampling rate. The coprime array, whose sparse structure and undersampling feature drastically decrease necessary computational and hardware cost, provides a theoretical foundation and technical basis for the increasing demands of its practical applications. Considering its superior performance in degrees-of-freedom, spatial resolution, and computational complexity, research on coprime array signal processing has attracted much attention. This paper reviews recent research progress on coprime array signal processing, which has focused on both the Direction-of-Arrival (DOA) estimation and adaptive beamforming. From the perspective of coprime array DOA estimation, this paper summarizes two typical approaches, namely the coprime subarray decomposition-based approach and the virtual array signal processing-based approach. Moreover, recent work on low-complexity and super-resolution DOA estimation via compressive sensing and gridless techniques is also introduced. From the perspective of coprime array adaptive beamforming, the differences and relationships between DOA estimation and beamforming in the framework of coprime array signal processing are discussed, and an efficient, robust, and adaptive beamformer design tailored for the coprime array is introduced. Advantages and the future directions of coprime array signal processing are discussed, along with the theoretical basis and a technical reference for practical radar applications.