We have investigated the heteroepitaxial growth and materials properties of pure and Co-doped TiO2 anatase on SrTiO3(001) and LaAlO3(001), grown by oxygen plasma assisted molecular beam epitaxy. This material is a promising new diluted magnetic semiconductor that shows large magnetization and a Curie temperature well above room temperature. We have found that epitaxial films with the highest crystalline quality and most uniform distribution of Co result when a rather slow growth rate ( similar to 0.01 nm/s) is used over a substrate temperature range of 550-600 degreesC on LaAlO3(001). These conditions result in layer-by-layer growth of single-crystal films and a very low density of extremely small nanocrystalline inclusions. In contrast, growth at a higher rate (similar to0.04 nm/s) leads to extensive formation of secondary-phase rutile nanocrystals to which Co diffuses and segregates. The rutile nanocrystals nucleate on the evolving anatase film surface in such a way that lattice strain between the two phases is ntinimized. Cobalt appears to substitute for Ti in the lattice and exhibits a +2 formal oxidation state. Both pure and Co-doped films can be grown as n-type semiconductors by controlled incorporation of oxygen atom vacancies. Free electrons are required to couple the Co(II) spin to a ferromagnetic state.