Excited-state theories in the single-reference, linear-response framework and their derivatives are reviewed with emphasis on their mutual relationship and applications to extended, periodic insulators. We derive configuration-interaction singles and time-dependent Hartree–Fock and perturbation corrections thereto including the so-called GW method. We discuss the accuracy and applicability of these methods to large molecules, in particular, excitons in crystalline polymers. We assess the potential of time-dependent density-functional theory (TDDFT) as an inexpensive, correlated excited-state theory applicable to large systems and solids. We list and analyze the weaknesses of TDDFT in calculating excitation energies and related properties such as ionization energies and polarizabilities. We also explore the equation-of-motion coupled-cluster hierarchy and low-order perturbation corrections. The issue of correct size dependence for an excited-state theory is addressed, relying on diagrammatic techniques.