In ordinary solids, shocks and cracks are extreme mechanical phenomena that require the application of extreme forces. But soft matter is different. Granular media, foams and polymer networks can be made so soft that even the tiniest perturbations elicit extreme mechanical responses. When that happens these materials are not just soft, they have become fragile. For fragile materials, the standard theoretical approach of linearizing the Hamiltonian fails. Their theoretical description requires a new approach based on non-linear waves and unusual modes of failure as the fundamental excitations of fragile matter. The ultimate aim of my research is to unveil how geometrical and topological features of the architecture of fragile materials control their extreme mechanics.