An elastic material, for that the stress tensor field of a certain deformation can be derived as gradient tensor field from a scalar potential function (called strain energy density function) is called a hyperelastic material. Hyperelasticity provides a means of modeling the non-linear stress–strain relationship of elastic polymers (elastomers). The first hyperelastic models have been developed by Ronald Rivlin and Melvin Mooney around 1950, since then many other (three dimensional) hyperelastic models have been developed. We want to introduce their classification and show examples, explain the difference in the modelling of compressible and incompressible materials, and discuss criterions that a strain energy density function should satisfy. For the modelling of elastic shells as two dimensional elastic surfaces we need good two dimensional hyperelastic models. For these, some adaptions need to be made.