As the smart grid evolves, it requires increasing distributed intelligence, optimization and control. Model predictive control (MPC) facilitates these functionalities for smart grid applications, namely: microgrids, smart buildings, ancillary services, industrial drives, electric vehicle charging, and distributed generation. Among these, this article focuses on providing a comprehensive review of the applications of MPC to the power electronic interfaces of distributed energy resources (DERs) for grid integration. In particular, the predictive control of power converters for wind energy conversion systems, solar photovoltaics, fuel cells and energy storage systems are covered in detail. The predictive control methods for grid-connected converters, artificial intelligence-based predictive control, open issues and future trends are also reviewed. The study highlights the potential of MPC to facilitate the high-performance, optimal power extraction and control of diverse sustainable grid-connected DERs. Furthermore, the study brings detailed structure to the artificial intelligence techniques that are beneficial to enhance performance, ease deployment and reduce computational burden of predictive control for power converters.