  The formation of polarons is a pervasive phenomenon in transition metal oxide
compounds, with a strong impact on the physical properties and functionalities
of the hosting materials. In its original formulation the polaron problem
considers a single charge carrier in a polar crystal interacting with its
surrounding lattice. Depending on the spatial extension of the polaron
quasiparticle, originating from the coupling between the excess charge and the
phonon field, one speaks of small or large polarons. This chapter discusses the
modeling of small polarons in real materials, with a particular focus on the
archetypal polaron material TiO2. After an introductory part, surveying the
fundamental theoretical and experimental aspects of the physics of polarons,
the chapter examines how to model small polarons using first principles schemes
in order to predict, understand and interpret a variety of polaron properties
in bulk phases and surfaces. Following the spirit of this handbook, different
types of computational procedures and prescriptions are presented with specific
instructions on the setup required to model polaron effects.
