Molecular beam epitaxy (MBE) is an important technique for the creation of new, non-equilibrium semiconductor materials and structures exhibiting novel physical phenomena. Surface diffusion plays an important role in the growth of these structures, influencing such fundamental growth processes and constants as islanding, critical thickness and epitaxial temperatures. Two approaches to the general problem of surface diffusion and islanding, using the SiGe system as a prototypical semiconductor heterostructure, are discussed: The time evolution of patterned deposits, and kinetic studies of nucleation and growth. While disordered laminar growth occurs for deposition at 300 K, elevated temperatures lead to Stranski-Krastanow (SK) growth (uniform coverage theta SK with excess Ge in islands). Diffusion coefficients for Ge on Si(100) have been determined for coverages below theta SK and show a significant coverage dependence. They are extremely sensitive to contamination with carbon on the order of approximately 0.05 ML, as well as to e-beam irradiation. In situ annealing experiments were performed to study the islanding process in real time. Provided the initial coverage exceeds the thickness of the SK layer, theta SK approximately 3 ML on Si(100)2 x 1, the initially uniform but disordered layer begins to collapse into a SK-type morphology at about 250 degrees C. At a ramping rate of 0.1 degrees C/s this process is completed at approximately 400 degrees C. A temperature dependence of the SK-layer thickness has been discovered for the first time. It is in excellent agreement with theoretical predictions.