Research efforts to bring forth epitaxial integration of elemental semiconductors (Si, Ge) and compound semiconductors (GaAs, InP, ZnSe, CdTe) in various combination thereof, are reviewed. Also reviewed are the issues relating to the growth of insulating layers on semiconductor layers for the purpose of forming SOI (semiconductor-on-insulator) for potential applications in 2D (two-dimensional) structures of 3D (3-dimensional) superstructures. First, the physics and chemistry of heteroepitaxial processes are examined along with the issues stemming from the mismatches of lattice, thermal and chemical origin. Ways to ease mismatches, using epitaxial control or intermediate layers like strained-layer-superlattice, are examined including recent progresses. Prospects for future directions along with technological implications of the semiconductor heterostructures are discussed, including a brief account of the history related to this technology.