This paper reviews progress in the heteroepitaxial growth of Ill-Nitride semiconductors. The growth of wurtzite and zinc-blende allotropic forms of GaN on various substrates with hexagonal and cubic symmetry respectively were discussed. In particular we addressed the growth on the various faces of sapphire, 6H-SiC and (001) Si. It has been shown that the kinetics of growth by plasma-MBE or ammonia-MBE are different. Specifically, in plasma-assisted MBE smooth films are obtained under group-III rich conditions of growth. On the other hand in ammonia-MBE smooth films are obtained under nitrogen rich conditions of growth. High quality films were obtained on 6H-SiC without the employment of any buffer. The various nucleation steps used to improve the two dimensional growth as well as to control the film polarity were discussed. The n- and p-doping of GaN were addressed. The concept of increasing the solubility of Mg in GaN by simultaneously bombarding the surface of the growing film with a flux of electrons (co-doping GaN with Mg and electrons) was discussed. The influence of the strength of Al-N, Ga-N and In-N bonds on the kinetics of growth of nitride alloys was pointed out. Specifically, it was shown that in both the nitrogen-rich and group-III rich growth regimes, the incorporation probability of aluminum is unity for the investigated temperature range of 750-800° C. On the other hand the incorporation probability of gallium is constant but less than unity only in the nitrogen-rich regime of growth. In the group-III regime the incorporation probability of gallium decreases monotonically with the total group-III flux, due to the competition with aluminum for the available active nitrogen. Alloy phenomena such as phase separation and atomic ordering and the influence of these phenomena to the optical properties were addressed. InGaN alloys are thermodynamically unstable against phase separation. At compositions above 30% they tend to undergo partial phase separation. Furthermore, InGaN alloys were found to undergo 1x1 monolayer cation ordering. AlGaN alloys do not show evidence of phase separation but they were found to undergo multiple type of superlattice ordering. Under nitrogen-rich growth conditions they show one monolayer periodicity, while under group-III rich growth it was found that the structure is a superposition of a seven monolayer and twelve monolayer superlattices. Finally, the growth of heterostructures and MQWs and the use of the MBE method for the fabrication of optical, electronic and electromechanical devices were discussed.