The issues of monitoring and fault diagnosis of drives with permanent magnet synchronous motors (PMSMs) are currently very relevant because of the increasing use of these drives in safety-critical devices. Every year, more and more articles are published on this subject. Therefore, the objective of this article is to update the overview of diagnostic methods and techniques for PMSM drives. Each of the main chapters of the article focuses on a specific element of the drive system (motor, power converter, measuring sensors), with particular emphasis on the components of the motor (stator windings, magnets, bearings, and rotor). The main sections on PMSM fault diagnosis are divided according to the type of methods used to obtain the symptoms of the damage. In addition, a review of methods that use the analysis of signals of the control structure for the diagnosis of damage to a vector-controlled motor is presented, as well as the latest achievements of researchers in the field of shallow and deep neural networks for the detection and classification of failures of PMSM drives. Based on the analyses presented in the literature, some development trends and challenges related to the development of diagnostics and fault-tolerant control of PMSM drives are discussed in the conclusion part.