Epitaxial silicon-germanium layers have already for some time been used in Si-based high performance devices. More recently integration of Ge layers in Si-based structures has been considered due to its superior carrier mobilities compared to silicon. The most feasible method for integrating SiGe and Ge layers into Si-based structures is through epitaxial growth. Among the epitaxial growth systems molecular-beam epitaxy (MBE) and ultra-high vacuum chemical-vapor deposition (UHV-CVD) are widely used, in particular in research environments, due to their flexibility with respect to controlling growth parameters. The emphasis of this review will be on structures grown by these two techniques. Due to the 4.2% lattice misfit between Si and Ge, SiGe and Ge layers on Si substrates are usually grown on a buffer layer to accommodate the strain, thus producing SiGe or Ge top layers of reduced defect densities. Top layers with or without strain can then be produced by adjusting the composition of the buffer relative to that of the top layer. A high quality top layer of low defect density is the required product, and different buffer-layer concepts on Si are discussed.