Epitaxial growth, a bottom-up self-assembly process for creating surface nano- and microstructures, has been extensively studied in the context of atoms. This process, however, is also a promising route to self-assembly of nanometer- and micrometer-scale particles into microstructures that have numerous technological applications. To determine whether atomic epitaxial growth laws are applicable to the epitaxy of larger particles with attractive interactions, we investigated the nucleation and growth dynamics of colloidal crystal films with single-particle resolution. We show quantitatively that colloidal epitaxy obeys the same two-dimensional island nucleation and growth laws that govern atomic epitaxy. However, we found that in colloidal epitaxy, step-edge and corner barriers that are responsible for film morphology have a diffusive origin. This diffusive mechanism suggests new routes toward controlling film morphology during epitaxy.