Materials with negative Poisson's ratio are characterized as auxetic materials. They arise quite rarely in nature, so that our engineering intuition can not help us understand their mechanical behaviour or, further, use them effectively for innovative products and processes. From a literature review and from analytical calculations confirmed with the use of numerical homogenization, it seems that some nonconvex-shaped (re-entrant) microstructure is the most understandable origin of the auxetic behaviour. The picture is quite clear in elastostatics, for which a quite large number of potential applications have been discussed in the engineering literature. Analogous problems in elastodynamics and in the area of viscoelasticity have not been discussed thoroughly in the open literature. The purpose of this article is to summarize the available knowledge in the area of auxetics and to point out interesting directions for further research and development work.