A critical torque detector is disclosed made up of a rotating input member, an output member supporting a drill and connected in a "flexible driving connection" with the input rotating member, and means for detecting difference in rotational speed between the input and output members due to a greater torsional resistance encountered by the drill during its operation than the critical torque predetermined for a specific drill. The input and output members are drivingly connected, as by a coil spring, magnetic repelling forces and so on, so as to rotate conjointly whenever a torsional resistance or a torque encountered by a running drill is lower than the predetermined value, but they are not always rigidly connected. Upon occurrence of greater torque imposed on the drill, the output member will be permitted to delay in rotation as compared to the input member. Such difference in rotational speed can be taken out mechanically or electromagnetically as a signal to detect occurrence of the critical torque.