Stall events in axial compressor systems have been a limiting factor for efficiency of such systems and a source of safety concerns. The detection of the onset of stall, and in many cases the precursor of the onset of stall, have been of interest in the axial compressor community for many decades. As such, development of algorithms along with active control could lower cost, reduce emissions, improve safety, and increase market competitiveness. To gain an understanding of these stall phenomena, past and current research has focused on modeling axial compressors as dynamic systems, with a focus on obtaining descriptive formulations of the physical aspects of stall. Some of these approaches allow for active control measures that extend the stall margin of the compressor system to increase safety and efficacy. This paper reviews the major contributions in these listed pursuits and presents the latest methods and algorithms for stall precursor detection in low-speed axial compressors. In particular, a review is presented in the types and characteristics of stalls, the major mathematical models used to describe these systems, influences of physical attributes such as tip clearance, guide vanes, and groove casing—operating as passive control elements—but also active control utilities such as air injection are discussed along with a detailed review of existing stall precursor detection algorithms. In addition, a forward-looking projection is presented that includes the use of machine learning algorithms to further the understanding and the capability of stall precursor detection.