The mechanisms of epitaxial growth affect both the form of a deposit and the way in which lattice defects are introduced into the deposit. These mechanisms are reviewed, with emphasis on the understanding of the growth mechanisms as determined from experimental observations, rather than theoretical treatments. The results of experimental observations discussed include those obtained by electron diffraction, electron microscopy, X-ray diffraction, scanning tunnelling microscopy, and atomic force microscopy. The structural details associated with the three well established growth modes, monolayer growth (Frank–van der Merwe), three-dimensional nucleation (Volmer–Weber), and monolayer growth followed by three-dimensional nucleation (Stranski–Krastanow), are discussed and summarised. Emphasis is given to growth by molecular beam epitaxy, because it allows the very early stages to be investigated. Examples are discussed to illustrate some of the main features of the details of the growth, and how they influence the overall structure of the deposits as growth proceeds. The aspects of the structure considered include the lattice spacings of the deposit (i.e. elastic strain), its physical form (e.g. isolated islands or continuous films), and its imperfection content, especially misfit dislocations. Under some conditions, lattice tilting of the deposit occurs.