We summarize information about materials with negative Poisson’s ratio (auxetic materials). We consider typical structures of natural and artificial auxetic materials. Employing the notion of similarity of mechanisms of deformation for various values of scale levels, we present a classification of these materials. It is shown that in most cases the negative sign of Poisson’s ratio is a consequence of particular modes of deformation of structural elements that constitute the material. We also consider materials such that their anomalous deformation is caused by peculiar properties of physical-chemical interactions and/or is the result of phase transformations. We present a model description of such anomalous behavior.