Atomically-thin, inherently 2D semiconductors offer thickness scaling of nanoelectronic devices and excellent response to light for low-power versatile applications. Using small exfoliated flakes, advanced devices and integrated circuits have already been realized, showing great potential to impact nanoelectronics. Here, high-quality single-crystal MoSe2 is grown by molecular beam epitaxy on AlN(0001)/Si(111), showing the potential for scaling up growth to low-cost, large-area substrates for mass production. The MoSe2 layers are epitaxially aligned with the aluminum nitride (AlN) lattice, showing a uniform, smooth surface and interfaces with no reaction or intermixing, and with sufficiently high band offsets. High-quality single-layer MoSe2 is obtained, with a direct gap evidenced by angle-resolved photoemission spectroscopy and further confirmed by Raman and intense room temperature photoluminescence. The successful growth of high-quality MoSe2/Bi2Se3 multilayers on AlN shows promise for novel devices exploiting the non-trivial topological properties of Bi2Se3.