Topological and geometrical properties of wave functions are becoming an indispensable tool with which to understand quantum phenomena in materials. These concepts are relevant to numerous fields including optics, electronics, metamaterials, superconductivity and quantum computation. However, the progress is somewhat biased and perhaps limited by the tempting assumption of perfect crystalline symmetry. In this talk I will give a pedagogical overview of how our group has extended and used these concepts to predict novel phenomenology in systems where crystalline symmetries are either absent, approximately satisfied, or only satisfied on average. I’ll use examples of new material platforms including amorphous solids, novel quasicrystals, and disordered crystals.