Detailed three-dimensional neutronics calculations have been performed for the U.S. design of the ITER magnet shield. The total nuclear heating in the TF coils is 35 kW in the technology phase and 42 kW in the physics phase. Using 5 cm thick W back shield layers behind the vacuum vessel in locations with limited shielding space results in acceptable local magnet damage levels. The parts of the TF coils adjacent to the divertor vacuum pumping ducts are well protected against streaming radiation.