Quantum geometry, which describes the geometry of quantum wave functions, plays a crucial role across a wide range of physics, including quantum information, subatomic physics, and cosmology. In particular, the quantum geometry of Bloch wave functions has become a cornerstone of modern quantum condensed matter physics. The quantum geometrical tensor encodes this geometry through two fundamental components: the quantum metric (the real part) and the Berry curvature (the imaginary part). While Berry curvature gained prominence through its manifestation in the intrinsic anomalous Hall effect, recent advances have revealed equally significant effects arising from the quantum metric. This includes its signatures in nonlinear transport, superfluid density of flat-band superconductors, and nonlinear optical responses. These advances underscore how quantum geometry is reshaping our understanding of condensed matter systems, with far-reaching implications for future technologies. This review surveys recent progress in the field, focusing on both foundational concepts and emergent phenomena in transport and optics—with a particular emphasis on the pivotal role of the quantum metric.

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• Antiferroelectricity
• Electrical properties
• Flat bands
• Valley degrees of freedom
• Topological materials