The paper discusses the problem of maximizing the torque of permanent magnet synchronous motors in the presence of voltage and current constraints. Formulas are given that are suitable to operation with voltage and current source inverters and to real-time computation. Generally, the available torque is limited by current constraints at low speeds and by voltage constraints at high speeds, but there is also often an intermediate range of speeds where both the voltage and the current constraints are active. The paper specifies how optimal operation can be achieved in all three ranges, and at what speeds the optimal operation transitions from one mode to another. A numerical example for a typical permanent magnet stepper motor is discussed to illustrate the contributions of the paper.