Heteroepitaxy allows the growth of semiconductor structures for a wide variety of device applications. When the materials are also lattice mismatched, the strain can be tailored to enhance the electronic properties of the active device layers. At the same time, however, misfit dislocations and other defects that tend to degrade device characteristics may be present. Strained Si MOSFETS are being developed for high-speed logic applications. These devices are built in a Si layer under biaxial tensile strain that is grown pseudomorphically on a so-called virtual substrate, typically a thick, strain-relaxed SiGe buffer layer on a Si(001) substrate. Key to the success of this application is the control of the misfit dislocations required to relieve the strain in the SiGe layer to achieve a low density of threading dislocations in the active device layers. Two types of SiGe/Si structures will be discussed. Strain-relaxed SiGe buffer layers produced by the implantation of He below the interface between a pseudomorphic SiGe layer and the Si(001) substrate and subsequent annealing are significantly more relaxed compared to layers of the same thickness that were not implanted. Platelet defects that are formed near the top of the Si substrate during annealing serve as dislocation nucleation sources. Elastic (defect free) strain relaxation of SiGe/Si structures is also under investigation. We have recently demonstrated that a pseudomorphic SiGe layer grown on free-standing Si relaxes elastically. The strain is shared with the free-standing Si layer resulting in Si under biaxial tensile strain.