Matter organizes in remarkable ways, giving us semiconductors, magnets, superconductors, and more exotic possibilities. I study the variety of behaviors possible, especially those that can only exist because of quantum mechanics and cannot be understood with classical pictures. Symmetry plays a key role, as in much of modern physics. I have been especially interested in topological band theory—such as topological insulators and Weyl semimetals—and topologically ordered systems such as quantum Hall effects and spin liquids.