to begin, i will describe a fast monte-carlo algorithm for simulating epitaxial surface growth, based on the continuous-time monte-carlo algorithm of bortz, kalos and lebowitz. when simulating realistic growth regimes, much computational time is consumed by the relatively fast dynamics of the adatoms. to solve this problem, we allow adatoms to take larger steps, determined by the local geometry, effectively reducing the number of transitions required. we achieve nearly a factor of ten speed up,for growth at moderate temperatures and moderate to low deposition rates. in the second part of the talk, i will discuss the computation of stained heteroepitaxial growth. elastic effects are incorporated using a ball and spring type model. an efficient method based on combining fourier and multigrid formulations is presented. the equations for the elastic displacement of atoms in the film are extended to a rectangular region by the use of fictitious atoms and a connectivity matrix, allowing the application of standard multigrid ideas. except for the top layer, the atoms in the substrate are completely removed and replaced by equivalent forces which can be efficiently evaluated using a fast fourier transform. this formulation has been implemented in both two and three dimensions. finally we introduce various approximations in the implementation of kmc to improve the computation speed. numerical results show that layer-by-layer growth is unstable if the misfit is large enough resulting in the formation of three dimensional islands.