Modular multilevel converter (MMC) is one of the most promising topologies for medium to high-voltage high-power applications. The main features of MMC are modularity, voltage and power scalability, fault tolerant and transformer-less operation, and high-quality output waveforms. Over the past few years, several research studies are conducted to address the technical challenges associated with the operation and control of the MMC. This paper presents the development of MMC circuit topologies and their mathematical models over the years. Also, the evolution and technical challenges of the classical and model predictive control methods are discussed. Finally, the MMC applications and their future trends are presented.