Atomistic Models -- Lattice Gas Models and Kinetic Monte Carlo Simulations of Epitaxial Growth -- Cluster Diffusion and Island Formation on fcc(111) Metal Surfaces Studied by Atomic Scale Computer Simulations -- A Multiscale Study of the Epitaxial CVD of Si from Chlorosilanes -- Off-lattice Kinetic Monte Carlo Simulations of Strained Heteroepitaxial Growth -- Quasicontinuum Monte Carlo Simulation of Multilayer Surface Growth -- Step Flow Models -- to Step Dynamics and Step Instabilities -- A Finite Element Framework for Burton-Cabrera-Frank Equation -- Edge Diffusion in Phase-Field Models for Epitaxial Growth -- Discretisation and Numerical Tests of a Diffuse-Interface Model with Ehrlich-Schwoebel Barrier -- Islands in the Stream: Electromigration-Driven Shape Evolution with Crystal Anisotropy -- Simulation of Ostwald Ripening in Homoepitaxy -- Continuum Models -- Continuum Models for Surface Growth -- Configurational Continuum Modelling of Crystalline Surface Evolution -- On Level Set Formulations for Anisotropic Mean Curvature Flow and Surface Diffusion.Epitaxy is a very active area of theoretical research since several years. It is experimentally well-explored and technologically relevant for thin film growth. Recently powerful numerical techniques in combination with a deep understanding of the physical and chemical phenomena during the growth process offer the possibility to link atomistic effects at the surface to the macroscopic morphology of the film. The goal of this book is to summarize recent developments in this field, with emphasis on multiscale approaches and numerical methods. It covers atomistic, step-flow, and continuum models and provides a compact overview of these approaches. It also serves as an introduction into this highly active interdisciplinary field of research for applied mathematicians, theoretical physicists and computational materials scientists.