A quasi-static mechanics analysis of nanoscale surface polishing that provides insight into the surface topography evolution and the removal of material at the asperity level is presented. The analysis is based on a three-dimensional stochastic model that accounts for multiscale (fractal) surface roughness and elastic, elastic-plastic, and fully plastic asperity deformation by hard abrasive nanoparticles embedded in the soft surface layer of a rigid polishing plate. Numerical results of the steady-state roughness of the polished surface, material removal rate, and wear coefficient are presented in terms of the apparent contact pressure, polishing speed, original topography and mechanical properties of the polished surface, average size and density of nanoparticles, and surface roughness of the polishing plate. Simulation trends are associated with elastic-plastic and fully plastic asperity contacts, responsible for irreversible topography changes (roughening effect) and material removal (smoothening effect), respectively. Analytical trends and predictions of the steady-state roughness of the polished surface and material removal rate are shown to be in good agreement with experimental results of nanoscale surface polishing (lapping) of magnetic recording ceramic heads.