  For many materials, a precise knowledge of their dispersion spectra is
insufficient to predict their ordered phases and physical responses. Instead,
these materials are classified by the geometrical and topological properties of
their wavefunctions. A key challenge is to identify and implement experiments
that probe or control these quantum properties. In this review, we describe
recent progress in this direction, focusing on nonlinear electromagnetic
responses that arise directly from quantum geometry and topology. We give an
overview of the field by discussing new theoretical ideas, groundbreaking
experiments, and the novel materials that drive them. We conclude by discussing
how these techniques can be combined with new device architectures to uncover,
probe, and ultimately control novel quantum phases with emergent topological
and correlated properties.
