Negative Poisson's ratio (NPR) or auxetic structures, including three-dimensional auxetic structures, are disclosed and applied to various applications. One such structure comprises a pyramid-shaped unit cell having four base points A, B, C, and D defining the corners of a square lying in a horizontal plane. Four stuffers of equal length or different lengths extend from a respective one of the base points to a point E spaced apart from the plane. Four tendons of equal length or different lengths, but less than that of the stuffers, extend from a respective one of the base points to a point F between point E and the plane. In three-dimensional configurations, a plurality of unit cells are arranged as tiles in the same horizontal plane with the base points of each cell connected to the base points of adjoining cells, thereby forming a horizontal layer. A plurality of horizontal layers are then stacked with each point E of cells in one horizontal layer being connected to a respective one of the points F of cells in an adjacent layer. Particularly for typical applications, the structure may further including a pair of parallel plates made sandwiching a plurality of horizontal layers of unit cells.