Materials and structures with negative Poisson’s ratio exhibit a counter-intuitive behaviour. Under uniaxial compression (tension), these materials and structures contract (expand) transversely. The materials and structures that possess this feature are also termed as ‘auxetics’. Many desirable properties resulting from this uncommon behaviour are reported. These superior properties offer auxetics broad potential applications in the fields of smart filters, sensors, medical devices and protective equipment. However, there are still challenging problems which impede a wider application of auxetic materials. This review paper mainly focuses on the relationships among structures, materials, properties and applications of auxetic metamaterials and structures. The previous works of auxetics are extensively reviewed, including different auxetic cellular models, naturally observed auxetic behaviour, different desirable properties of auxetics, and potential applications. In particular, metallic auxetic materials and a methodology for generating 3D metallic auxetic materials are reviewed in details. Although most of the literature mentions that auxetic materials possess superior properties, very few types of auxetic materials have been fabricated and implemented for practical applications. Here, the challenges and future work on the topic of auxetics are also presented to inspire prospective research work. This review article covers the most recent progress of auxetic metamaterials and auxetic structures. More importantly, several drawbacks of auxetics are also presented to caution researchers in the future study.