This chapter gives an introduction to the epitaxial growth of thin films on solid substrates. The term epitaxy refers to the growth of a crystalline layer on (epi) the surface of a crystalline substrate, where the crystallographic orientation of the substrate surface imposes a crystalline order (taxis) onto the thin film. This implies that film elements can be grown, up to a certain thickness, in crystal structures differing from their bulk. If film and substrate have the same crystal lattices, but different lattice constants, the film will be under strain, that is, it will have a slightly different lattice constant than in its own bulk. Both effects, together with the electronic hybridization at the interface, lead to novel properties.One distinguishes homo-and heteroexpitaxy, where the former refers to the growth on one element on a crystal surface of its own and the latter refers to the more general case, where film and substrate materials are different. Note that the first distinguishes itself from crystal growth, as we will see in more detail later.We start this chapter by giving examples from technology, illustrating where thin epitaxial films are used and outlining potential applications that become reality once we are able to grow the respective thin film sequences. We then contrast thin film and crystal growth, respectively, with kinetics and thermodynamics of growth. We introduce the deposition techniques used in epitaxial thin film growth and then discuss the classical thermodynamic approach, which led to the definition of the growth modes. These modes refer to the morphology taken on by a system grown close to thermodynamic equilibrium. Often films are grown far away from equilibrium and their morphology is determined by kinetics, that is, it is the result of the microscopic path taken by the system during growth. This path is determined by the hierarchy of rates of the single atom, cluster or molecular precursor displacements as compared to the deposition rate. Owing to the importance of kinetics, we focus in the rest of the chapter on the kinetic description of growth. In order to simplify the topic, we start with coverages below one atomic layer that is referred to as a monolayer. The first submonolayer part will be on nucleation, followed by a discussion of island shapes that, very much like snowflakes, tell us about the elementary processes that took place during their formation. We then discuss island coarsening, either by evaporation of atoms from their edges, referred to as the Ostwald ripening, or by the diffusion and subsequent coalescence of entire islands, referred to as the