Patrick Rinke Fritz-Haber-Institut der Max-Planck-Gesellschaft Theory Excited states are ubiquitous. They can be of direct technological interest for example in optoelectronic devices or be an intermediary in experimental techniques that probe the response of a system to external perturbations such as light or electrons. In this lecture, I will focus on single-particle-like excitations or quasiparticles and explore the connections to band structures and photoelectron spectra. I will introduce Green’s function theory and the GW approximation (where G is the single particle Green’s function and W the screened Coulomb interaction) as a practical tool to compute single-particle excitations.