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BiFeO3 thin films have been deposited on (001) SrTiO3, (101) DyScO3, (011) DyScO3, (0001) AlGaN/GaN, and (0001) 6H-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 in accordance with thermodynamic calculations. Four-circle x-ray diffraction and transmission electron microscopy reveal phase-pure, epitaxial films with rocking curve full width at half maximum values as narrow as 7.2 arc seconds (0.002deg). Epitaxial growth of (0001)-oriented BiFeO3 thin films on (0001) GaN, including AlGaN HEMT structures, and (0001) SiC has been realized using intervening epitaxial (111) SrTiO3 / (100) TiO2 buffer layers. The epitaxial BiFeO3 thin films have 2 in-plane orientations:  BiFeO3 || [112macr0] GaN (SiC) plus a twin variant related by a 180deg in-plane rotation. This epitaxial integration of the ferroelectric with the highest known polarization, BiFeO3, with high bandgap semiconductors is an important step toward novel field-effect devices.