Model predictive control (MPC), manly based on a direct use of an explicit and identifiable model, has been widely used in controller design in different applications both by academia and industry. The reason for such popularity is due to its strong ability for providing high performance electric drive systems, as highly recognized as the most reliable control approach compared with field-oriented control (FOC) and direct torque control (DTC). In general, the MPC has numerous features and advantages, such as direct switching states to the converter without any modulation, online optimization with multivariable control, low current total harmonic distortion, low switching loss, etc. The aim of this paper is to provide a comprehensive review for major development and achievements of the recent progress on the MPC for electrical machines and drives. This review begins with the innovative topologies and operating principles of fundamental MPC, and ends to summary on different advanced MPC algorithms. Typical MPC techniques have been fully adopted to enhance the drive performance of the electrical drives, mainly including finite-set model predictive control (FS-MPC) based on tuning weighting factors, without weighting factors, maximum torque per ampere, low number of switching vectors, and multi voltage vectors in one sample period. Finally, great attention has been paid to the discussion of the new trends and future research topics.