*RECRUITMENT TALK* Epitaxial growth is a modern technology to grow thin solid films by depositing atoms or molecules onto an existing layer of material. Microscopic processes in epitaxial growth include the deposition of atoms onto a surface, desorption of adsorbed atoms (adatoms) into the gas phase, surface diffusion of adatoms, attachment and detachment of adatoms to and from atomic steps, adatom island nucleation, and island coalescence. These processes are characterized by fluctuation, non-equilibrium, and multiple spatial and temporal scales. In this talk, I will first present a derivation of an island dynamics model for epitaxial growth that includes step kinetics. This is an improvement of the classical Burton-Cabrera-Frank model that assumes the adatom equilibrium along steps. An adaptive finite element method for the new model will be described. I will then focus on a class of fourth-order diffusion equations that model the coarsening in the surface dynamics of growing films after the roughening transition. Such growth equations are gradient flows of certain free energies. I will show bounds for coarsening rates and the decay of energy, derive the energy asymptotics in the large-system-limit, and predict the exact scaling laws for the coarsening under the hypothesis of realization.