  The rate of second layer nucleation -- the formation of a stable nucleus on
top of a two-dimensional island -- determines both the conditions for
layer-by-layer growth, and the size of the top terrace of multilayer mounds in
three-dimensional homoepitaxial growth. It was recently shown that conventional
mean field nucleation theory overestimates the rate of second layer nucleation
by a factor that is proportional to the number of times a given site is visited
by an adatom during its residence time on the island. In the presence of strong
step edge barriers this factor can be large, leading to a substantial error in
previous attempts to experimentally determine barrier energies from the onset
of second layer nucleation. In the first part of the paper simple analytic
estimates of second layer nucleation rates based on a comparison of the
relevant time scales will be reviewed. In the main part the theory of second
layer nucleation is applied to the growth of multilayer mounds in the presence
of strong but finite step edge barriers. The shape of the mounds is obtained by
numerical integration of the deterministic evolution of island boundaries,
supplemented by a rule for nucleation in the top layer. For thick films the
shape converges to a simple scaling solution. The scaling function is
parametrized by the coverage $\theta_c$ of the top layer, and takes the form of
an inverse error function cut off at $\theta_c$. The surface width of a film of
thickness $d$ is $\sqrt{(1- \theta_c) d}$. Finally, we show that the scaling
solution can be derived also from a continuum growth equation.
