The earliest ideas of the polaron recognized that the coupling of an electron to ionic vibrations would affect its apparent mass and could effectively immobilize the carrier (self-trapping). We discuss how these basic ideas have been generalized to recognize new materials and new phenomena. First, there is an interplay between self-trapping and trapping associated with defects or with fluctuations in an amorphous solid. In high dielectric constant oxides, like HfO2, this leads to oxygen vacancies having as many as five charge states. In colossal magnetoresistance manganites, this interplay makes possible the scanning tunnelling microscopy ( STM) observation of polarons. Second, excitons can self-trap and, by doing so, localize energy in ways that can modify the material properties. Third, new materials introduce new features, with polaron-related ideas emerging for uranium dioxide, gate dielectric oxides, Jahn-Teller systems, semiconducting polymers and biological systems. The phonon modes that initiate self-trapping can be quite different from the longitudinal optic modes usually assumed to dominate. Fourth, there are new phenomena, like possible magnetism in simple oxides, or with the evolution of short-lived polarons, like muons or excitons. The central idea remains that of a particle whose properties are modified by polarizing or deforming its host solid, sometimes profoundly. However, some of the simpler standard assumptions can give a limited, indeed misleading, description of real systems, with qualitative inconsistencies. We discuss representative cases for which theory and experiment can be compared in detail.