In this thesis, we are dealing with modelling and numerical treatment of epitaxial growth of thin crystalline films, where we concentrate on diffuse-interface approximations of two descriptions of this process. In the first part, we consider a semi-continuous model resolving atomic distances in the growth direction but being coarse grained in the lateral directions. Mathematically, this leads to a free boundary problem proposed by Burton, Cabrera and Frank for steps separating terraces of different atomic heights coupled to a diffusion equation for the adatom (adsorbed atom) concentration fulfilling certain boundary conditions at the steps. For this sharp-interface model, a diffuse-interface approximation can be given by a viscous Cahn-Hilliard equation, which is based on a Ginzburg-Landau free energy. Great emphasis is put on the incorporation of an Ehrlich-Schwoebel barrier — a higher energy barrier for attachment to a step down, which leads to a jump in the adatom concentration at the steps — as well as diffusion along step edges and anisotropic effects into a diffuse-interface model. We provide a justification by matched asymptotic expansions formally showing the convergence of the diffuse-interface model towards the sharp-interface model as the interface width shrinks to zero. The numerical treatment of the viscous Cahn-Hilliard is based on a semi-implicit finite element discretization, where an adaptive strategy of local mesh refinement and coarsening has been applied. Computational results include the numerical reproduction of the results of the asymptotic analysis in one dimensional situations, the investigation of the stability of a circular island and simulations of anisotropic island growth and spiral growth. The second model is continuous in all directions. We thereby assume that the interface between the film and the vapour is represented by a smooth surface, whose evolution is given by a geometric law that combines surface diffusion and interface kinetics, which can again be approximated by a viscous Cahn-Hilliard equation. In principle, we reuse the previous numerical approach and validate it with an investigation of an instability caused by an additional elastic energy. Further examples show the smoothing property for closed curves and surfaces as well as the evolution towards anisotropic shapes.