Permanent-magnet synchronous motors (PMSMs) have become the preferred choice in the industrial sector owing to their high power density, excellent efficiency, and wide speed control range. Over the past decade, advancements in high-performance drives and power electronics have propelled the development of model predictive control (MPC) technology. Among the various approaches, finite-control-set MPC (FCS-MPC) has attracted significant interest for the control of permanent-magnet motors owing to its simple structure and fast response. Three key challenges in FCS-MPC are explored by leveraging recent research advancements: selecting appropriate weighting factors in the cost function, optimizing vector selection, minimizing computation time in multi-vector control, and improving system robustness under parameter mismatches. Additionally, the evolution and application of FCS-MPC in PMSMs are discussed.