Magnetic shields and magnetic shielding systems are described. The excessive spatial demands of known mu-metal/cryoperm and superconducting shielding systems are reduced by a new multi-piece shield construction approach. A complete magnetic shielding system for use with superconducting-based computing systems, such as superconducting quantum computing systems, is also described. This complete system may include mu-metal/cryoperm shields and superconducting shields using either compensatory magnetic fields, expulsion by temperature gradients, or a combination of the two.