This paper is focused mainly on the subject of elastic magnetic electron scattering and how it has come to be a useful tool for studying the spatial distributions of convection and magnetization currents in the nuclear ground state. Using such a probe, emphasis is clearly placed on the electromagnetic currents provided by the valence nucleons in the nucleus, indeed, by neutrons as well as protons. These do not yield a complete description of the problem, however, and more complex nuclear many-body configurations are generally required. Various models are employed (shell-model configuration mixing, core-polarization effects, models of deformed nuclei, for instance), giving rise to quantitative comparisons with experimental measurements. Moreover, to achieve such successful descriptions, especially at high momentum transfers, it is frequently necessary to go beyond a purely one-body nucleonic reaction mechanism and to include the effects of two-body meson-exchange currents based on non-nucleonic degrees of freedom ($\pi$,$\rho$,$\Delta ,\dots$). The authors discuss all of these various facets of the problem, beginning with surveys of the historical development of the field and of the experimental techniques employed in such studies. They present a detailed treatment of the formalism needed in discussions of elastic electron scattering, including an introduction to the density-matrix approach to the nuclear many-body problem, with simple examples to clarify the ideas involved. For the nonspecialist a separate section of illustrative examples is supplied in which qualitative discussions of various aspects of the physics accessible in such ($e$,$e^{\prime }$) studies are highlighted. Experimental and theoretical results for a large number of nuclei ranging from $A=2\mathrm{} \mathrm{to} 209$ are described in detail; this represents a compilation of virtually all of the high-quality data that are available at present. The paper concludes with projections as to which directions may be followed in the future, in particular, with a relatively complete discussion of the use of polarization in elastic electron scattering.

DOI:https://doi.org/10.1103/RevModPhys.56.461