In this paper we review our recent work developing the growth of AlGaN/GaN high-electron mobility transistors (HEMTs) grown on SiC (0001) by plasma-assisted molecular beam epitaxy (PA-MBE). State-of-the-art AlGaN/GaN HEMTs have been achieved using MBE-grown material. Buffer leakage was an important limiting factor for early devices. We have shown that by appropriately controlling the Al/N flux ratio during growth of the nucleation layer on SiC(0001), low-leakage GaN buffers can be subsequently grown. In addition, a "modulated growth" technique was developed to achieve large area uniformity and surface morphology control. High-performance HEMTs were fabricated utilizing these two techniques. On 200 nm gate-length devices, at 4 GHz an output power density of 8.4 W/mm was obtained with a power-added efficiency (PAE) of 67% at a drain bias of 30 V. At a higher drain bias (42 V), 13.7 W/mm with a PAE of 55% was achieved.