  We investigate the kinetics of submonolayer epitaxial growth which is driven
by a fixed flux of monomers onto a substrate. Adatoms diffuse on the surface,
leading to irreversible aggregation of islands. We also account for the
effective diffusion of islands, which originates from hopping processes of
their constituent adatoms, on the kinetics. When the diffusivity of an island
of mass k scales as k^{-mu}, the (mean-field) Smoluchowski rate equations
predicts steady behavior for 0<=mu<1, with the concentration c_k of islands of
mass k varying as k^{-(3-mu)/2}. For mu>=1, a quasi-static approximation to the
rate equations predicts slow continuous evolution in which the island density
increases as ln t^{mu/2}. A more refined matched asymptotic expansion reveals
unusual multiple-scale mass dependence for the island size distribution. Our
theory also describes basic features of epitaxial growth in a more faithful
model of growing circular islands. For epitaxial growth in an initial
population of monomers and no external flux, a scaling approach predicts
power-law island growth and a mass distribution with a behavior distinct from
that of the non-zero flux system. Finally, we extend our results to one- and
two-dimensional substrates. The physically-relevant latter case exhibits only
logarithmic corrections compared to the mean-field predictions.
