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This thesis deals with the nonlinear aspects of superconducting quantum interference devices (SQUIDs) as magnetic meta-atoms. Such meta-atoms are usually resonant structures that constitute the basic building blocks of a metamaterial with the purpose of giving the material unconventional magnetic properties. Due to their intrinsic nonlinearity, SQUIDs exhibit a number of phenomena such as frequency tunability and multi-stability that make them attractive as controllable meta-atoms.