  Quantum geometry has been identified as an important ingredient for the
physics of quantum materials and especially of flat-band systems, such as
moir\'e materials. On the other hand, the coupling between light and matter is
of key importance across disciplines and especially for Floquet and cavity
engineering of solids. Here we present fundamental relations between
light-matter coupling and quantum geometry of Bloch wave functions, with a
particular focus on flat-band and moir\'e materials, in which the quenching of
the electronic kinetic energy could allow one to reach the limit of strong
light-matter coupling more easily than in highly dispersive systems. We show
that, despite the fact that flat bands have vanishing band velocities and
curvatures, light couples to them via geometric contributions. Specifically,
the intra-band quantum metric allows diamagnetic coupling inside a flat band;
the inter-band Berry connection governs dipole matrix elements between flat and
dispersive bands. We illustrate these effects in two representative model
systems: (i) a sawtooth quantum chain with a single flat band, and (ii) a
tight-binding model for twisted bilayer graphene. For (i) we highlight the
importance of quantum geometry by demonstrating a nonvanishing diamagnetic
light-matter coupling inside the flat band. For (ii) we explore the twist-angle
dependence of various light-matter coupling matrix elements. Furthermore, at
the magic angle corresponding to almost flat bands, we show a
Floquet-topological gap opening under irradiation with circularly polarized
light despite the nearly vanishing Fermi velocity. We discuss how these
findings provide fundamental design principles and tools for
light-matter-coupling-based control of emergent electronic properties in
flat-band and moir\'e materials.
