A model for surface growth with some of the features of molecular-beam epitaxy is proposed and investigated. Particles are deposited randomly on a one-dimensional substrate and the surface relaxes through diffusion processes, which obey detailed balance. The model undergoes a phase transition from a rough phase to a grooved phase. Both phases display scaling in space and time, with equal exponents. We also propose a Langevin equation which should describe this growth process and show that this equation contains an infinite number of relevant terms.