Crystalline metal oxides display a large number of physical functionalities such as ferroelectricity, magnetism, superconductivity, and Mott transitions. High quality heterostructures involving metal oxides and workhorse semiconductors such as silicon have the potential to open new directions in electronic device design that harness these degrees of freedom for computation or information storage. This review describes how first-principles theoretical modeling has informed current understanding of the growth mechanisms and resulting interfacial structures of crystalline, coherent, and epitaxial metal oxide thin films on semiconductors. Two overarching themes in this general area are addressed. First, the initial steps of oxide growth involve careful preparation of the semiconductor surface to guard against amorphous oxide formation and to create an ordered template for epitaxy. The methods by which this is achieved are reviewed, and possibilities for improving present processes to enable the epitaxial growth of a wider set of oxides are discussed. Second, once a heterointerface is created, the precise interfacial chemical composition and atomic structure is difficult to determine unambiguously from experiment or theory alone. The current understanding of the structure and properties of complex oxide/semiconductor heterostructures is reviewed, and the main challenges to prediction-namely, (i) are these heterostructures in thermodynamic equilibrium or kinetically trapped, and (ii) how do the interfaces modify or couple to the degrees of freedom in the oxide?-are explored in detail for two metal oxide thin films on silicon. Finally, an outlook of where theoretical efforts in this field may be headed in the near future is provided.