This review deals with the structure of hadrons, strongly interacting many-body systems consisting of quarks and gluons. These systems have a size of about $1\mathrm{fm}$, which shows up in scattering experiments at low momentum transfers $Q$ in the GeV region. At this scale the running coupling constant of quantum chromodynamics (QCD), the established theory of the strong interactions, becomes divergent. It is therefore highly intriguing to explore this theory in the realm of its strong interaction regime. However, the quarks and gluons cannot be resolved at the GeV scale but have to be studied through their manifestations in the bound many-body systems, for instance, pions, nucleons, and their resonances. The review starts with an overview of QCD at low momentum transfer and a summary of the theoretical apparatus describing the interaction of hadrons with electrons and photons. Experimental results are presented for the most significant observables studied with the electromagnetic probe: form factors, polarizabilities, excitation spectra, and sum rules. These experimental findings are compared and interpreted with various theoretical approaches to QCD, such as phenomenological models with quarks and pions, dispersion relations as a means to connect observables from different experiments, and, directly based on the QCD Lagrangian, chiral perturbation theory and lattice gauge theory.

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