A robotic bending apparatus for bending archwires and other types of elongate, bendable medical devices into a desired configuration includes a first gripping tool and a moveable gripping tool. The first gripping tool can be either fixed with respect to a base or table for the robot or positioned at the end of robot am. The moveable gripping tool is mounted to the end of a moveable robot arm having a proximal portion also mounted to the base. The robot preferably comprises a six axis bending robot, in which the distal end of the moveable arm can move relative to the fixed gripping tool about three translational axes and three rotational axes. The gripping tools preferably incorporate force sensors which are used to determine overbends needed to get the desired final shape of the archwire. The robot may also include a resistive heating system in which current flows through the wire while the wire is held in a bent condition to heat the wire and thereby retain the bent shape of the wire. A magazine for holding a plurality of straight archwires needing to be bent and a conveyor system for receiving the wires after the bending process is complete are also described. The robot bending system is able to form archwires with any required second and third order bends quickly and with high precision. As such, it is highly suitable for use in a precision appliance-manufacturing center manufacturing a large number of archwires (or other medical devices or appliances) for a distributed base of clinics.