It is generally observed that strain relaxation, which occurs by misfit dislocation formation, in lattice-mismatched heteroepitaxial layers is accompanied by the formation of threading dislocations. However, our group and others have observed that strain-relaxed epitaxial layers of In1−xGaxAs on GaAs substrates can be grown without the formation of threading dislocations in the epitaxial layer. We have been able to grow strain-relaxed layers up to 13% In concentration without observable densities of threading dislocations in the epilayer but do observe a large number of dislocations pushed into the GaAs substrate. The ability to grow strain-relaxed, lattice-mismatched heteroepitaxial layers has important practical applications. We have succeeded in growing dislocation-free layers of ZnSe on appropriately lattice-matched layers of In1−xGaxAs.