To address difficulties in radar signal processing, the effective and efficient detection of lowobservable moving targets in complex environments is an ongoing research hotspot. On the one hand, a signal may be extremely weak due to strong clutter and the complex motion of a target, making it hard to separate them in the time and frequency domains. On the other hand, complex coherent integration methods and the heavy computational burden of long-time integration represent challenges for improving radar detection performance with limited resources. High-resolution sparse representation can separate clutter from a moving target with respect to signal sparsity, and can be regarded as an extension of traditional transform-based moving target detection methods. This method has promising application prospects due to the advantages of its high time-frequency resolution, anti-noise property, robustness, and suitability for the analysis of multi-signals. In this paper, we systematically review conventional radar moving target detection methods. Then, we summarize their applications, including sparse representation in clutter property analysis, suppression, moving target detection, signature extraction, and time-frequency analysis. Next, we consider future developments. Finally, we provide some results based on real datasets and existing research.