Hydrogen (H2) is a new type of renewable energy that can meet people's growing energy needs and is environmentally friendly. In order to improve the industrial application prospects and electrochemical performance of hydrogen evolution catalysts, extensive research on transition metal materials has been carried out. Among the many catalytic materials, cobalt is an element with potential for the hydrogen evolution reaction (HER) due to its abundant reserves, low cost, and small energy barrier for H adsorption. This review classifies the latest research on cobalt-based catalysts according to the types of compound, including cobalt-based sulfides, phosphides, carbides, borides, oxides, etc., and summarizes the latest research progress of cobalt-based compound catalysts in acidic and alkaline media. Strategies to tune the properties of cobalt-based compound catalysts for high catalytic activity for HER are focused on, including structural engineering, defect engineering, and doping, etc. The advantages and limitations of each modified approach are reviewed. Not only that, but also the catalytic activity and advantages of the catalyst are evaluated by using density functional theory (DFT) calculation-related descriptors, activity evaluation parameters, etc. Finally, limitations and challenges of cobalt-based materials for HER are presented, as well as prospects for future research. This paper aims to understand the chemical and physical factors that affect cobalt-based catalysts, and to find directions for future research on cobalt-based catalysts.