InN is an important III-V compound semiconductor with many potential microelectronic and optoelectronic applications. In this study, we prepared epitaxial InN on (0001) sapphire with an AlN buffer layer by molecular beam epitaxy, and its variation, migration enhanced epitaxy. A series of samples were grown with different substrate temperatures ranging from 360°C to 590°C. The optimum growth temperature for InN was found to be between 450°C and 500°C. We also found that thicker AlN buffer layers result in the best InN quality. With increasing thickness of an AlN buffer layer, the Hall electron mobility of InN increases while the carrier concentration decreases. The surface morphology is also improved this way. Hall mobility greater than 800 cm2/Vs with carrier concentration 2-3×1018 cm−3 at room temperature can be routinely obtained for ∼0.1[.proportional]m thick InN films. Various InN-based heterostructures with AlInN or AlN barrier were fabricated. X-ray diffraction study clearly shows the barrier and InN layers. A 2-dimensional electron gas resulting from polarization induced electrons was observed in capacitance-voltage measurements. Some results on Mg doping of InN will be discussed as well.