To reduce failure and enhance performance, it would be desirable to predict and control the residual stress in thin films. This requires a fundamental understanding of how the stress is related to the underlying processes controlling film growth. We describe the results of rate equation models and computer simulations focused on advancing this understanding. The rate equation models explore the competition between different stress-inducing mechanisms occurring simultaneously during growth. Tensile stress is induced by the interaction between islands as they merge to form new segments of grain boundary (island coalescence). At the same time, compressive stress is induced by diffusion of atoms into the grain boundary driven by the non-equilibrium conditions on the surface. The stress can be further modified by densification of the film due to grain growth. The model predictions for the dependence of the stress on growth rate, diffusivity and grain size are compared with measurements for systems with different types of microstructural evolution. To understand how the surface kinetics are related to the stress, we have also developed kinetic Monte Carlo simulations that include similar mechanisms for generating stress at the grain boundaries. This approach enables us to look at the stochastic effect of individual atom dynamics on the stress evolution. For instance, we can study how supersaturation on the surface due to the deposition flux acts as a driving force for compressive stress. Results for the evolution of stress at different growth rates, temperatures and grain sizes will be shown.