Understanding new superconductors requires high-quality epitaxial thin films to explore intrinsic electromagnetic properties and evaluate device applications(1-9). So far, superconducting properties of ferropnictide thin films seem compromised by imperfect epitaxial growth and poor connectivity of the superconducting phase(10-14). Here we report new template engineering using single-crystal intermediate layers of (001) SrTiO3 and BaTiO3 grown on various perovskite substrates that enables genuine epitaxial films of Co-doped BaFe2As2 with a high transition temperature (T-c,T-rho=0 of 21.5 K, where rho = resistivity), a small transition width (Delta T-c = 1.3K), a superior critical current density J(c) of 4.5 MA cm(-2) (4.2 K) and strong c-axis flux pinning. Implementing SrTiO3 or BaTiO3 templates to match the alkaline-earth layer in the Ba-122 with the alkaline-earth/oxygen layer in the templates opens new avenues for epitaxial growth of ferropnictides on multifunctional single-crystal substrates. Beyond superconductors, it provides a framework for growing heteroepitaxial intermetallic compounds on various substrates by matching interfacial layers between templates and thin-film overlayers.