Abstract: Auxetic materials and structures have special mechanical properties. In contrast to traditional materials, auxetic materials contract (expand) under uniaxial compression (tension). Auxetic materials have many desirable properties and are superior to traditional materials in terms of shear capacity, fracture resistance, energy absorption capacity and indentation resistance. Because of these excellent properties, auxetic materials are very promising in the fields of medical equipment, intelligent filters, intelligent sensors, protective equipment, aviation, navigation and national defense engineering, but the application and popularization of auxetic materials still face some challenges. In this paper, the domestic and foreign research results of auxetic materials are extensively reviewed and the latest progress of auxetic materials is introduced. Auxetic materials are roughly classified into the following four categories: natural auxetic materials, cellular auxetic materials, metallic auxetic materials, multi-material auxetics and auxetic composites. Internal structure, auxetic deformation mechanism and mechanical properties of each kind are discussed. Some excellent mechanical properties of auxetic materials are summarized and new inventions and applications of auxetic materials in various industries are introduced as well. In view of the fact that the actual application of auxetic materials are much less than the research theories and experimental results, this paper analyzes some shortcomings and the current challenges of auxetic materials, including high manufacturing cost, high porosity, insufficient bearing capacity and only suitable for small strain conditions. To promote the application of auxetics, the metallic auxetic materials are introduced in detail. The steps and methods for designing and fabricating 3D metallic auxetic materials are proposed as well.