Infrared detection systems have the characteristics of good concealment, strong anti-jamming ability, etc. and are widely applied in military and civil fields. The detection of small and weak targets is an important part of an infrared detection system and has become an attractive research area. Recently, scholars have made remarkable achievements in the research of infrared dim small target detection algorithms based on the low-rank sparse decomposition. This study focuses on the research status and development of infrared dim small target detection algorithms based on the low-rank sparse decomposition and presents a detailed review on three aspects: background component constraints, target component constraints, and joint time-domain information constraints. First, the constraints of the background component are divided into the low-rank constraint of block image, low-rank constraint of tensor, and full variation constraints. Second, the constraints of the target component are analyzed from two aspects: the sparse representation of targets and the target component weighting strategy of fusing local priors. Then, the joint time-domain information constraint is analyzed. Furthermore, the performances of a typical detection algorithm based on the low-rank sparse decomposition and a single frame detection algorithm are compared. Finally, future research direction in this field is highlighted.