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