A number of discrete models as well as continuum equations have been proposed for describing epitaxial and thin film growth. We have shown that there exists a macroscopic groove instability in many of these models. This unphysical feature in the continuum equations arises from the truncation or linearization of the diffusion operator along the surface. A similar artifact occurs in the discrete models, because in these models adatoms only diffuse horizontally and must take an unphysical vertical jump at step edges. We have proposed and studied a continuum equation for epitaxial and thin-film growth in which the full diffusion along the surface is taken into account. The results of the solutions of this continuum equation, for the growth and the morphology of the surface, are in excellent agreement with recent low temperature molecular-beam epitaxy and ion-sputtering experiments. In particular, we find that at late times dynamic scaling breaks down and the surface is no longer a self-affine fractal. The surface develops a characteristic morphology whose dependence on deposition rate and surface diffusion is similar to that found in experiments.
