As the modern industry rapidly advances toward digitalization, networking, and intelligence, intelligent fault diagnosis technology has become a necessary measure to ensure the safe and stable operation of mechanical equipment and effectively avoid major disaster accidents and huge economic losses caused by mechanical equipment failure. As the “power heart” of hydraulic transmission systems, hydraulic piston pumps (HPPs) occupy an important position in aerospace, navigation, national defense, industry, and many other high-tech fields due to their high-rated pressure, compact structure, high efficiency, convenient flow regulation, and other advantages. Faults in HPPs can create serious hazards. In this paper, the research on fault recognition technology for HPPs is reviewed. Firstly, the existing fault diagnosis methods are described, and the typical fault types and mechanisms of HPPs are introduced. Then, the current research achievements regarding fault diagnosis in HPPs are summarized based on three aspects: the traditional intelligent fault diagnosis method, the modern intelligent fault diagnosis method, and the combined intelligent fault diagnosis method. Finally, the future development trend of fault identification methods for HPPs is discussed and summarized. This work provides a reference for developing intelligent, efficient, and accurate fault recognition methods for HPPs. Moreover, this review will help to increase the safety, stability, and reliability of HPPs and promote the implementation of hydraulic transmission technology in the era of intelligent operation and maintenance.