The ITER magnet system, which is composed of toroidal field (TF) and poloidal field (PF) coils, central solenoid (CS), correction coils, and all their structural supports, is cooled by supercritical helium flow at 4.5 K. Eddy currents are induced in the metallic components during normal operation, due to the variation of PF and CS coils current, and to the plasma ramp-up and ramp-down. In addition, during disruptions, as well as other fast plasma transient events, the eddy currents circulating in these structures reach very high values due to the high induced electric fields. An effective cooling is therefore needed to limit the temperature increase of the magnets and their supports. The Joule energy dissipated in the cold structures of the ITER magnet system has been computed by means of the electromagnetic finite-element code CARIDDI. A model of a 40 degree sector of the ITER magnet has been built in order to represent in detail the connections between the CS and the TF coils, PF and CS supports, and all the intercoil structures and the metallic part of the TF coils (radial plates, case, etc.). Vacuum vessel, thermal shield, and cryostat have also been modeled. The reference 15-MA inductive scenario (with plasma current ramp-up duration of 80 s and ramp-down of 200 s) and a 15-MA fast inductive scenario characterized by the fastest plasma current ramp-up (50 s) and the fastest plasma current ramp-down (65 s) have been analyzed. In addition, several plasma instabilities have been simulated considering the reference electrical connections between the metallic parts.