We have studied the epitaxy of few-layer hexagonal boron nitride (h-BN) by plasma-assisted molecular beam epitaxy (MBE) using a low growth rate and nitrogen-rich condition. It has been determined that under such conditions, the growth temperature is the factor having the most significant impact on the structural and optical quality of the material. When grown at temperatures <1000 °C, the h-BN film is polycrystalline, and defect-related photoluminescence (PL) emission dominates. Epitaxial domains of exceptional crystalline quality are obtained at elevated substrate temperatures of ~1300 °C, which exhibit strong band-edge PL emission at ~220 nm and negligible defect-related emission at room temperature. Our atomistic calculations reveal that, even though the gap of h-BN is indirect, it luminesces as strongly as direct-gap materials. Experimentally, the luminescence intensity of such a few-layer h-BN sample is measured to be two orders of magnitude stronger than that of a 4-µm thick commercially grown AlN template on sapphire, demonstrating the extraordinary potential of epitaxial h-BN for deep ultraviolet (UV) optoelectronics and quantum photonics.