Graphene, as a typical representative of the two-dimensional material family, has received a wide attention due to its excellent physical and chemical properties. Graphene nanoribbon (GNR) is graphene in a width of several to a few tens of nanometers. GNRs not only inherit most of the excellent properties of graphene, but also have their own specific properties such as band gap opening and spin-polarized edge states, which make it the potential candidate in graphene based electronics in the future. Hexagonal boron nitride (h-BN), which has similar lattice constant with graphene, normally serves as an ideal substrate for graphene and GNRs. It can not only effectively preserve their intrinsic properties, but also benefit for the fabrication of electrical devices via popular semiconductor processes. In this paper, we reviewed the development history of research of graphene and GNRs on h-BN in recent years. The recent progress of physical properties is also discussed. In order to realize the large scale production of graphene and GNRs on h-BN, high quality h-BN multilayer is necessary. In addition, recent progresses about h-BN preparation methods are presented, and the progresses could pave the way for the further application of GNRs in the electronics. Finally, the research direction of graphene and GNRs on h-BN in the future is discussed.