As an ultrawide band gap semiconductor, two-dimensional (2D) hexagonal boron nitride (h-BN) has attracted considerable research interest because of its unique properties such as excellent electrical insulation, high breakdown field, high thermal conductivity, and good chemical inertness, which make it a promising candidate as dielectric and substrate layers for devices based on other 2D materials. The synthesis of high-quality and large-area h-BN layers with few defects is strongly desirable for these applications. In this review, the methods and progress of the epitaxial growth of 2D h-BN on transition metal, h-BN films on dielectric and semiconductor substrates are described in detail. The high-quality 2D h-BN layers can be epitaxially grown on the transition metal substrates with catalytic activity including Cu, Ni, Fe, Pt, etc. However, it is a great challenge to directly grow h-BN single crystal films on insulating dielectric or semiconductor substrates. Sapphire is a preferred substrate for the growth of h-BN for its good thermal stability and chemical stability. The growth of h-BN thin films on sapphire substrate has been widely reported by various techniques, such as chemical vapor deposition, molecular beam epitaxy, ion beam sputtering deposition, metal-organic vapor phase epitaxy, high-temperature post-annealing and so on. Based on these techniques, high-quality 2D h-BN has been prepared on sapphire substrates, it can also be integrated into the epitaxial growth processes of some existing III-V compound semiconductors, laying the foundation for the large-scale application of h-BN. In addition, the growth of h-BN single crystal films have also been attempted on semiconductor substrates like graphene, silicon and germanium, which provide an attractive strategy for the fabrication and application of h-BN-based heterojunctions.