A review of the growth of oriented oxides on Si and Ill-V semiconductors provides insight into some of the common themes of oxide/semiconductor epitaxy. The nature and success of the epitaxy can be attributed to four primary factors: (i) semiconductor surface preparation; (ii) oxide/semiconductor reaction thermodynamics; (iii) surface and interfacial polarity; and (iv) structural matching (lattice matching, thermal expansion, and symmetry). Semiconductor surface preparation governs the initial stages of epitaxy for systems such as MgO/GaAs and In2O3/InAs. In these cases, the epitaxial development depends on the presence or absence of a native oxide layer prior to growth. Chemical reaction can also influence the epitaxial process, as is illustrated in the growth of gadolinium oxide on Si. In general, the initial stages of epitaxy reflect a thermodynamic competition between the formation of the desired oxide phase, oxidation of the semiconductor, and formation of intermediate phases such as silicides and silicates. An analysis of possible reactions is presented for selected binary and ternary oxides with Si and GaAs. Surface and interfacial energy can also play an important role in determining the morphology and orientation of oxides having polar low-index faces, as illustrated in the growth of fluorite and related bixbyite oxides such as CeO2, In2O3 and Y2O3. The epitaxial relationships between the oxide and semiconductor may be rationalized in terms of either direct lattice matching or higher order epitaxy.