Epitaxial thin films of pure-phase Fe3O4(110), Fe3O4(111), alpha-Fe2O3(11 (2) over bar 0), and alpha-Fe2O3(1 (1) over bar 02) have been grown on MgO(110), alpha-Al2O3(0001), alpha-Al2O3(11 (2) over bar 0), and alpha-Al2O3(1 (1) over bar 02) substrates, respectively, by molecular beam epitaxy using an elemental Fe source and an electron cyclotron resonance oxygen plasma source. Characterization of the crystal structures, chemical states, and epitaxial relationships was carried out using a variety of techniques, including in situ reflection high-energy electron diffraction (RHEED), low-energy electron diffraction, x-ray photoelectron spectroscopy/diffraction, and ex situ x-ray reflectivity and diffraction. Real-time RHEED reveals that Fe3O4 growth on MgO appears in a step-flow fashion, whereas the growth of Fe3O4(111) on alpha-Al2O3(0001) occurs initially by island formation, and then island coalescence. However, the growth of alpha-Fe2O3 on alpha-Al2O3 appears to follow an intermediate growth mode. The formation of pure-phase films is controlled largely by oxygen partial pressure, plasma power, and growth rate, but appears to be independent of growth temperature, at least from 250 to 550 degrees C. The present study demonstrates that selective growth of pure-phase iron oxides with various low-index orientations can be achieved by controlling the growth conditions and selecting suitable substrates.