Stents of the type used to treat and prevent localized flow constriction in body vessels are based upon negative Poisson's ratio (NPR) structures. An auxetic stent constructed in accordance with this invention comprises a tubular structure having two ends defining a length with a central longitudinal axis and an axial view defining a cross section. The tubular structure is composed of a plurality of unit cells with two different configurations, called V-type and X-type. In V-type auxetic stents, each unit cell comprises a pair of side points A and B defining a width, a first pair of members interconnecting points A and B and intersecting at a point C forming a first V shape, and a second pair of members interconnecting points A and B and intersecting at a point D forming a second V shape. In X-type auxetic stents, each unit cell comprises eight points from A to H defining an outline of the unit cell. Eight straight or curved members interconnecting points A and B, B and C, C and D, C and E, E and F, F and G, G and H, G and A, respectively, forming the X-type unit cell. In both configurations, the unit cells are connected in rows and columns, such that compression of the structure between the two ends thereof causes the cross section of the structure to shrink in size. The auxetic structure configurations invented can also be used, with similar dimensions or significantly different dimensions, for other applications, such as in a nano-structural device, a tubal fastener design, or in an application associated with a large oil pipe or other pipelines.