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Epitaxial thin film growth by vapor deposition or molecular beam epitaxy under ultra‐high vacuum conditions generally occurs in two stages: (i) nucleation and growth of well‐separated islands on the substrate; (ii) subsequent formation of a thicker continuous film with possible kinetic roughening. For homoepitaxial growth, two‐dimensional (2D) monolayer islands are formed during submonolayer deposition. Typically, the presence of a step‐edge barrier inhibits downward transport and leads to the formation of mounds (multilayer stacks of 2D islands) during multilayer growth. For heteroepitaxial growth, islands formed in the initial stages of deposition sometimes have a 2D monolayer structure. However, they may instead exhibit bilayer or 3D multilayer structure due to, e.g., a high film surface energy, strain, or quantum size effects. Various growth modes are possible for thicker films. Atomistic modeling provides the most detailed picture of film growth. For coherent (defect‐free) epitaxial films, lattice‐gas modeling analyzed by kinetic Monte Carlo simulation (KMC) is particularly successful in describing film growth on the appropriate time and length scales. For large islands or complex systems, another effective and instructive approach is laterally coarse‐grained step‐dynamics modeling which tracks only the evolution of step edges in each layer. However, fully coarse‐grained 3D continuum modeling for the evolution of a film height function does not yet have predictive capability. Examples are provided for: Ag homoepitaxy on (100), (111) and (110) surfaces; Ag heteroepitaxy on lattice‐matched substrates including NiAl(110), NiAl(100), and Fe(100); and Ag heteroepitaxy on 5‐fold icosohedral Al‐Pd‐Mn and 2‐fold decagonal Al‐Cu‐Co quasicrystalline surfaces.