BiFeO3 thin films have been deposited on (101) DyScO 3, (0001) AlGaN/GaN, and (0001) SiC single crystal substrates by reactive molecular-beam epitaxy in an adsorption-controlled growth regime. This is achieved by supplying a bismuth over-pressure and utilizing the differential vapor pressures between bismuth oxides and BiFeO3 to control stoichiometry. Four-circle x-ray diffraction reveals phase-pure, epitaxial films with rocking curve full width at half maximum values as narrow as 7.2 arc seconds (0.002°). Epitaxial growth of (0001)-oriented BiFeO3 thin films on (0001) GaN, including AlGaN HEMT structures, and (0001) SiC has been realized utilizing intervening epitaxial (111) SrTiO3 / (100) TiO2 buffer layers. The epitaxial BiFeO3 thin films have two in-plane orientations: [112̄0] BiFeO3 || [112̄0] GaN (SiC) plus a twin variant related by a 180° in-plane rotation. This epitaxial integration of the ferroelectric with the highest known polarization, BiFeO3, with wide band gap semiconductors is an important step toward novel field-effect devices.