This review brings together experimental data on surface shape evolution during epitaxial growth of GaAs with kinetic Monte Carlo simulations of a solid-on-solid model and numerical solutions of a continuum growth equation derived from an adatorn transport equation. Scanning probe and light scattering measurements of the surface morphology of GaAs, grown by molecular beam epitaxy, on planar as well as patterned (100) substrates are reviewed. We show that the experimental data can be described by a stable continuum growth equation that is mixed-order in the spatial derivatives, with an Edwards-Wilkinson type linear term, together with a conservative nonlinear term. The stable growth equation is derived from two coupled rate equations, one of which describes the transport of adatoms on the surface and the other describes the rate of change of surface height due to adatorn incorporation into the surface at step edges. In this analysis, we assume that there is a combination of an Ehrlich-Schwoebel barrier and/or an incorporation barrier at step edges that favor a net downhill migration of adatoms across step edges, with the consequence that the growth model, like the experimental system, is stable, meaning that undulations in the surface tend to smooth out during growth. The coefficients in the growth equation depend on the growth rate and the density of steps on the surface. The continuum description of the morphological evolution is tested by comparisons to computer experiments consisting of kinetic Monte Carlo simulations of a solid-on-solid model. The methods used in this analysis of GaAs epitaxy are expected to be broadly applicable to other materials that exhibit stable epitaxial growth. (c) 2008 Published by Elsevier B.V.