A plasma-arc cutting apparatus having a plasma torch producing a plasma arc and cutting a workpiece with the plasma arc, the plasma torch including a nozzle from whcih the plasma arc is ejected toward the workpiece; a feed device moving the plasma torch relative to the workpiece; a magnetic-field forming device producing a magnetic filed, and applying the magnetic field to the plasma arc so as to intersect the plasma arc and thereby deflect the arc; a first control device numerically controlling the feed device according to cutting operation information, the cutting operation information including cutting path data representative of a predetermined curved cutting path along which the plasma torch is moved over the workpiece, the first control means providing torch position data representative of a current position of the plasma torch relative to the workpiece; and a second control device controlling the magnetic-field forming device according to the cutting operation information and the torch position data so that the plasma arc is deflected to an optimum direction with respect to a direction of movement from the current position of the plasma torch as the plasma torch is moved along the curved cutting path.