Using a simple mechanical device the principles of complex hysteresis are explained and demonstrated experimentally. Subsequently it is shown, how these principles find ubiquitous applications for various kinds of systems. Examples are magnetic materials, shape memory alloys, or the adsorption-desorption process in porous materials. Basic methods for the modeling of such systems are introduced. The interaction between systems showing complex hysteresis and their environment leads to novel challenges in the field of nonlinear dynamics. Some aspects thereof are highlighted.