Transmission electron microscopy is a technique capable of determining specimen composition and structure at atomic resolution. Since the invention of the first microscopes in the 1930’s, a series of instrumentation and theoretical advances have led to a mature field where one can routinely obtain images of individual columns of atoms with high precision. Despite the extraordinary success of the microscope and its prevalence in laboratories around the world, interpretation of experimental images is a non-trivial matter. The strong interaction of electrons with the charged particles making up the target specimen (~10^4 times greater than that for x-rays) results in interesting and complicated scattering dynamics which can make direct interpretation of experimental images difficult. A particularly important scattering mechanism at high incident energies is thermal scattering, whereby the incident electron excites a phonon (that is, a vibrational mode) within the specimen. In this thesis we will present a new model for thermal scattering which affords new physical insights as compared with previous models. In particular the new model distinguishes between elastic and thermal scattering of the fast electron and can predict the individual contributions to the scattered intensity from both types of scattering. We will use this model to gain new insights into the role of thermal scattering in a number of different imaging modes.