Electric motors act as the backbone of industrial development. Their reliable and safe operation is essential to various industries. At present, motor fault diagnosis based on current signatures has progressively gained favour thanks to their non-invasive nature. This review summarizes recent advances in motor current signature analysis for fault diagnosis. First, motor diagnostic background and requirements are expounded, as well as the benefits of current signature analysis. Then, the mechanism and influence of common faults for the most widely used induction motor and permanent magnet synchronous motor are analyzed, and the detection criteria for typical motor defects, such as bearing fault, stator windings inter turn short circuit, and broken rotor bar, are summarized. Next, the motor diagnosis techniques based on current signature analysis are summarized from the technical perspectives of spectrum analysis, demodulation transformation, time-frequency analysis, parameter estimation, artificial intelligence, and etc. Pros and cons of each technology are also provided. Finally, according to the challenges faced by the existing technologies in engineering, future research is suggested to focus on influence analysis of electromechanical coupling on current signal, feature extraction for incipient fault, non-stationary signal analysis, unlabeled data utilization and fault severity assessment.