A Monte-Carlo-based method for simulation of epitaxial crystal growth is reported. The improvement of this concept in comparison with the simulations based on the solid-on-solid model is the possibility of simulating growth without the assumption of a discrete crystal lattice structure. The used continuous space model requires only the assessment of the Hamiltonian. In the first step the model was applied for studying growth in a two-dimensional cross-section through the substrate and the epilayer with a width of up to 80 atoms. Two different potentials can be chosen for the simulation : (a) Lennard-Jones and (b) directional Lennard-Jones to describe covalent systems. In addition to the selection of the potential, we can set the following parameters for different kinds of atoms : radii of the potential minimum, binding energy between the atoms for all possible pairings, growth temperature and substrate structure. Island and layer growth in the nucleation stage were investigated by selecting different binding energies of the epilayer atoms to the substrate atoms. The epitaxial growth of strained layers was simulated to study the incorporation of dislocations at the interface. The continuous space model also allowed calculation of the rhombic distortion of a strained layer grown on a tilted substrate with atomic steps. As a further extension of the two-dimensional model the re-evaporation of particles was allowed which is essential for the simulation of atomic-layer epitaxy. The obtained results for the growth rate in atomic layer epitaxy as a function of substrate temperature can be compared with experimental findings. In a second attempt we extended the continuous space model to three dimensions based on the same assumption as tested in the two-dimensional model. The three-dimensional model allowed the same simulations for islands and layer growth but also the selection of vicinal surfaces of the substrates with respect to the usually used (111)-oriented surfaces. The simulation showed a clear stabilisation of the growth direction by the atomic steps on the substrate and even in lattice-matched structures the formation of twins was suppressed.