Model predictive control (MPC) has emerged as a promising approach to control a modular multilevel converter (MMC). With the help of a cost function, the control objectives of an MMC are achieved easily by using an MPC approach. However, the MPC has several technical challenges and issues including the need of accurate system models, computational complexity, and variable switching frequency operation and weighting factor selection, when it comes to the control of an MMC. In the past few years, several research studies are conducted to address some of the challenges and issues in an MPC and developed several model predictive algorithms for an MMC. In this paper, the importance of each challenge and its impact on the system performance is discussed. Also, the MMC mathematical models used in the implementation of MPC are presented. Furthermore, some of the popular MPC algorithms are discussed briefly, and their features and performance are highlighted through case studies. Finally, summary and future trends of MPC for an MMC are presented.