Positive displacement compressors are essential in many engineering systems, from domestic to industrial applications. Many studies have been devoted to providing more insights into the workings and proposing solutions for performance improvements of these machines. This study aims to present a systematic review of published research on positive displacement compressors of various geometrical structures. This paper discusses the literature on compressor topics, including leakage, heat transfer, friction and lubrication, valve dynamics, port characteristics, and capacity control strategies. Moreover, the current status of the application of machine learning methods in positive displacement compressors is also discussed. The challenges and opportunities for future work are presented at the end of the paper.