In materials science, auxetic behavior refers to the rather unusual property of a material to laterally expand, rather than shrink, when stretched in some direction. Only sporadic examples are known. I will present a brief survey of the geometric theory of auxetic behavior for periodic bar-and-joint frameworks, as introduced in a series of recent papers by Ciprian Borcea and myself. The theory leads to new principles for designing an abundance of frameworks that provably support auxetic deformations. It also leads to effective algorithms for deciding when a given framework allows, infinitesimally, an auxetic deformation. We have developed code to effectively test for the auxetic property on frameworks built from crystal databases. I will conclude with a summary of the current challenges and some preliminary results. This is joint work with Ciprian Borcea (Rider U.) and Juan Castillo (Harvard U.). Our project was supported by a 2018-19 Harvard Radcliffe IAS Fellowship.