Negative Poisson's ratio (NPR) or auxetic are used to make lightweight wheels and runflat tires. The NPR tires can be tailored and functionally-designed to optimally meet the runflat requirements for both military and commercial vehicles. NPR-runflat tires may be fabricated using standard materials and simple manufacturing processes, resulting in low-cost and high-volume production. In preferred embodiments the runflat tire designs are fully compatible with Central Tire Inflation Systems (CTIS), while providing a performance equivalent to current military vehicle solutions but at half the weight. An auxetic wheel according to the invention comprises a line defining an axis of rotation; and a plurality of concentric rings of unit cells surrounding the axis, each unit cell being constructed of a plurality of members defining a Negative Poisson's Ratio (NPR) structure. The outermost ring of unit cells is arranged to facilitate rolling terrain contact, such that the stiffness of the structure in the localized region of loading due to terrain contact increases as the wheel rotates. A layer of material may be disposed between the concentric rings of unit cells which in preferred embodiments comprise a plurality of nested-V shapes. A cover may be provided over the outermost ring of unit cells forming a tire which may, or may not, be inflated.