Machining apparatus for machining a workpiece with a machining tool, including a work holder for holding the workpiece, and a cold gas stream supply device including a cooling gas nozzle through which a cold gas stream is supplied to a machining point, and wherein the cooling gas nozzle is provided on the work holder, and the cold gas stream has a temperature lower than an ambient temperature of the apparatus. The apparatus may be a centerless grinding apparatus including a regulating wheel for rotating the workpiece, and a work rest blade cooperating with the regulating wheel to rotatably support the workpiece so that the workpiece is ground by a grinding wheel while the workpiece is held by the work holder between the regulating and grinding wheels. The cooling gas nozzle is desirably positioned such that a distance between the exit end of the cooling gas nozzle and the grinding point is not larger than a diameter of the workpiece. Also disclosed are machining process and cooling method and device using the cold gas stream.