We discuss spectral properties of layer potential operators arising in electrostatic problems. In particular, we review known results and present some recent and ongoing developments concerning the spectra of the single-layer potential operator and the Neumann–Poincaré operator. We then establish Weyl-type laws for layer potential operators, which describe the asymptotic behavior of their eigenvalues and their relation to the geometry of the underlying domain.
 As an application, we discuss surface plasmon phenomena, which have attracted considerable attention from physicists and applied mathematicians, and explain that such phenomena do not occur for strictly convex smooth bounded domains in three dimensions. We also find a related result showing that cloaking by anomalous localized resonance does not occur for strictly convex bounded domains with smooth boundaries in three dimensions.
 Finally, we introduce open problems concerning how the geometry and regularity of the boundary affect the spectral behavior of layer potential operators.