This chapter lays down the foundation for elastic stability of columns, circular plates, rectangular plates, cylindrical shells and spherical shells that possess negative Poisson’s ratio. Results show that the plate Poisson’s ratio has no effect on the elastic stability of rectangular plates under in-plane biaxial loadings when the critical buckling load is expressed in terms of plate flexural rigidity, but the Poisson’s ratio plays a greater role for circular plate buckling. In the elastic stability study of spherical shells, the critical buckling stress is directly proportional to the shell thickness for Poisson’s ratio of 0 and proportional to the square of the shell thickness as the Poisson’s ratio approaches −1. Thereafter a summary of results by Miller et al. (Compos Sci Technol 70:1049–1056, 2010
) for flatwise buckling optimization of hexachiral and tetrachiral honeycombs is furnished. Finally examples are given for square plates with array of perforations such that imposition of uniaxial compressive buckling leads to 2D auxeticity.