We develop a global fast terminal sliding mode (GFTSM)-based model predictive current control (MPCC) for a three-phase permanent magnet synchronous motor (PMSM) drive system. To overcome load disturbance and parameter variation, we design a GFTSM-based speed regulator, and prove its stability and robustness. To reduce stator current ripples and improve control accuracy, we present the MPCC method to select the optimal voltage vector of a voltage source inverter. The resultant GFTSM-based MPCC PMSM drive system not only runs reliably and steadily but also has a satisfactory control effect for its torque and speed. Compared with two MPCC PMSM drive systems, namely, the PI-based regulator and the conventional sliding mode-based regulator, the GFTSM-based regulator possesses better dynamical performance and stronger robustness, as well as weaker chattering and a smaller total harmonic distortion index of three-phase stator currents. Simulation results validate the feasibility and effectiveness of the proposed scheme.