  The interaction of electrons with crystal lattice vibrations (phonons) and
collective charge-density fluctuations (plasmons) influences profoundly the
spectral properties of solids revealed by photoemission spectroscopy
experiments. Photoemission satellites, for instance, are a prototypical example
of quantum emergent behavior that may result from the strong coupling of
electronic states to plasmons and phonons. The existence of these spectral
features has been verified over energy scales spanning several orders of
magnitude (from 50 meV to 15-20 eV) and for a broad class of compounds such as
simple metals, semiconductors, and highly-doped oxides. During the past few
years the cumulant expansion approach, alongside with the GW approximation and
the theory of electron-phonon and electron-plasmon coupling in solids, has
evolved into a predictive and quantitatively accurate approach for the
description of the spectral signatures of electron-boson coupling entirely from
first principles, and it has thus become the state-of-the-art theoretical tool
for the description of these phenomena. In this chapter we introduce the
fundamental concepts needed to interpret plasmon and phonon satellites in
photoelectron spectra, and we review recent progress on first-principles
calculations of these features using the cumulant expansion method.
