  We present models of surfaces of crystals in an environment where molecular
beam epitaxy (MBE) and related methods of crystal growth like atomic layer
epitaxy (ALE) can be performed. Besides detailed models of reconstructed (001)
surfaces of II-VI semiconductors like CdTe and ZnSe, we study generic models to
adress fundamental properties of epitaxial growth like kinetic roughening and
the coarsening of mounds.
  A lattice gas model of flat (001) surfaces of CdTe and ZnSe in thermal
equilibrium suggest an interpretation of the transition between a c(2x2)
reconstruction at low temperature and a (2x1) reconstruction at high
temperature as an encompanying effect of an order-disorder phase transition. We
extend this lattice gas model to models of three-dimensional crystals. Using
one of these models, we have performed the (to the best of our knowledge) first
simulations of ALE.
  We apply the WTMM method to investigate the scaling properties of kinetically
rough surfaces obtained by the simulation of MBE growth on the surface of a
simple cubic crystal. Our results support a generalization of Family-Vicsek
scaling. We show, that this is the most general scaling behaviour if dynamic
scale invariance is fulfilled and a correlation length is the only relevant
lengthscale of the surface.
  Finally, we consider the coarsening of mounds on surfaces of crystals having
different lattice structures. We find, contrary to the results of a continuum
theory of Siegert et. al. and Golubovic et. al., that the dynamic exponents are
independent of the surface symmetry.
