Techniques and ideas from discrete, computational and algebraic geometry, rigidity theory, matroids and graph theory can be brought together to shed new light on problems inspired by crystallography and materials science. Here, the objects of study are molecules and crystals modeled as mechanical frameworks: periodic geometric graphs subject to distance constraints. The central theme of this presentation is a coherent and abundantly illustrated theory of geometric auxetics, along with efficient algorithms, procedures for designing new structures and open problems. This is joint work with Ciprian Borcea.