A model is presented to describe the evolution of thin-film surface morphology during growth and ion erosion. Characteristic in-plane length scales and overall amplitude of the roughness are studied as a function of certain competing roughening and smoothing mechanisms. Particular attention is paid to the deposition method of growth followed by ion erosion of an excess layer thickness. The model is extended to the case of multilayers, to include roughness correlations between different interfaces. Specular and diffuse x-ray scattering measurements on Mo/Si multilayers are interpreted in terms of the model. Quantitative agreement between the model and the experimental data can be obtained if we assume viscous flow to be the dominant smoothing mechanism during ion erosion of the Si layers.