The auxeticity of elastic materials is described and explained by the use of the linear and nonlinear models of elastic deformations for a wide range of strain values up to moderate levels. This chapter consists of three parts – general information on auxetic materials, description of auxetics by the model of the linear theory of elasticity, description of auxetics by the models of the nonlinear theory of elasticity. The analytical expressions are offered that corresponds to three kinds of universal deformations (simple shear, uniaxial tension, omniaxial tension) within the framework of three well-known in the nonlinear theory of elasticity models – two-constant Neo-Hookean model, three-constant Mooney-Rivlin model, five-constant Murna-ghan model. A most interesting novelty consists in that the sample from elastic material is deformed as the conventional material for small values of strains whereas as the auxetic with increasing to moderate values of strains.