Developments in design and technology in the engineering and medical fields necessitate the use of smart and high-performance materials to meet higher engineering specifications. The general requirements of such materials include a combination of high stiffness and strength with significant weight savings, resistance to corrosion, chemical resistance, low maintenance, and reduced costs. Over the last three decades, it has been demonstrated that auxetic materials offer a huge potential for the fields of engineering, natural sciences, and biomedical engineering, and for many other industries, including the aerospace and defense industries, through their unique deformation mechanism and measured enhancements in mechanical properties. To meet future engineering challenges, auxetic materials are increasingly being recognized as integral components of smart and advanced materials. Although materials with a negative Poisson’s ratio have been known since the early 1900s, they did not capture researchers’ attention until the late 1980s. Since 1991, these materials have been known as auxetic materials. Since then, their benefits and applications have been expanded to all major classes of materials such as metals, ceramics, polymers, and composites, and they are also now being used in engineering applications. The goal of this review was to present the development of auxetic polymers, which were first fabricated in the form of polyurethane foam approximately three decades ago and are now used in the fabrication of non-woven nano/micropolymeric structures. This review could provide useful information for the future development of auxetic polymers.