To provide a clutch device or a brake device of a press machine capable of surely and smoothly moving a clutch disc or a brake disc without sliding resistance with a simple mechanism.

This clutch device has the clutch disc for performing clutch operation by being sandwiched between a clutch rotary plate and a pressing plate for operating a piston by being supported by an end surface of a flywheel, or the brake device has the brake disc for performing brake operation by being sandwiched between a brake rotary plate and the pressing plate for operating the piston by being supported by a frame of the press machine. The clutch disc 1 or the brake disc is supported by the flywheel, a crankshaft or the frame of the press machine by using a plate-like spring 6 or an annular plate-like spring.