We report on the growth Of high electron mobility AlGaN/GaN heterostructures on sapphire substrates by gas-source molecular beam epitaxy (GSMBE) using ammonia as the nitrogen source. Improvements in structural, electrical, and optical properties of GaN and AlGaN layers have been made to achieve this goal. For the growth of AlGaN layers, the reflection high-energy electron diffraction revealed a twofold surface reconstruction, indicative of atomic smoothness of the film surface. High mobility two-dimensional electron gas has been achieved in both unintentionally doped (by piezoelectric effect induced by lattice mismatch strain) and modulation doped AlGaN/ GaN heterostructures. The modulation-doped n+-Al0.2Ga0.8N/i-GaN heterojunction exhibited electron mobilities as high as 750 and 4070 cm(2)/V s at 300 and 77 K, respectively. Both values are the highest ever reported for the AlGaN/GaN heterostructures grown by MBE techniques.