Advances in the ability to fabricate complex structures stands to revolutionize our understanding of how a material’s mechanical response is imbued by its geometry. This has led to the idea of a “mechanical metamaterial,” a designed structure whose mechanical properties are substantially different than those of the stuff from which it is made. By exploring the mechanics of disordered and ordered networks of springs, we have been trying to understand the deep connection between elastic response and geometry. I will tell you about recent theoretical work in which spring networks qualitatively change their mechanical response (from completely floppy to rigid) by small changes in their conformations. These kinds of critical states turn out to be ubiquitous and governed by a hidden topological structure we have only partially understood.