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

Topology and geometry under the nonlinear electromagnetic spotlight