Direct graphene growth on semiconductor substrates is an important goal for successful integration of graphene with the existing semiconductor technology. We test the feasibility of this goal by using molecular beam epitaxy on various semiconductor substrates: group IV (Si, SiC), group III-V (GaAs, GaN, InP), and group II-VI (ZnSe, ZnO). Graphitic carbon has been formed on most substrates except Si. In general, the crystallinities of carbon layers are better on substrates of hexagonal symmetry than those on cubic substrates. The flatness of graphitic carbon grown by molecular beam epitaxy is noticeable, which may help the integration with semiconductor structures.