Extensive kinetic Monte Carlo simulations have been used to model the growth of epitaxial films by molecular beam epitaxy (MBE). We allow both deposition and diffusion (single particle hopping events) to occur simultaneously and have investigated several different rules for hopping. Simple solid-on-solid models have been used with L x L substrate sizes and with nearest neighbor interactions only. Interfacial width and reflection high energy electron diffraction (RHEED) intensity have been determined as measures of the roughness of the surface. We find that in the long time limit the interfacial width has a logarithmic dependence on time and substrate size, although there is a pronounced crossover region for moderate times which depends on both temperature and flux. These results suggest that these models fall in the universality class of the Edwards-Wilkinson model. Comparison with experimental results shows similar behavior up to intermediate times and suggests that experimental data are needed over longer time scales.