The damping function has been a concept introduced in rheology since more than 30 years ago, and although a similar concept was already earlier implemented in studying rubber materials, its implementation in the modeling of polymer melts was an essential step forward in the classification and understanding of nonlinear viscoelasticity phenomena. It is the objective of this contribution to give an overview on the theoretical background and physical interpretation of the concept of the damping function for different types of deformation, as well as a review on the experimental results including the experimental artefacts to be considered. Besides homopolymers, a summary is given on different investigations of other types of systems, where the concept of the damping function has also been applied, for example, rubbers, rubber-like materials, block copolymers, polymer composites, liquid crystals, polymer blends, suspensions, emulsions, micellar systems, and in food rheology.