Using a combination of atomistic and continuum modeling, we describe various features observed during heteroepitaxial growth of Ge/Si thin films in a molecular beam epitaxy unit. At submonolayer coverage, dimer chain formation is observed, resulting from surface reconstruction. Rebonding in dimer chains leads to stabilization of vacancy structures around 1 monolayer (ML) coverage. In the 3ML-8ML coverage, the surface shows 3-dimensional island formation, which is nucleation limited for high strain films. For low strain films, island formation is governed by instability of the surface and is suitably modeled using continuum methods. Through these studies, we have achieved a thorough understanding of heteroepitaxial growth. We have recently extended our continuum model to describe growth on patterned substrates. In this case, the wavelength of the pattern competes with the wavelength of the instability to produce rich phase behavior, wherein the islands can move to different locations on the pattern.