  We report on the observation of high-temperature ($T_\textrm{c}$)
superconductivity and magnetic vortices in single-unit-cell FeSe films on
anatase TiO$_2$(001) substrate by using scanning tunneling microscopy. A
systematic study and engineering of interfacial properties has clarified the
essential roles of substrate in realizing the high-$T_\textrm{c}$
superconductivity, probably via interface-induced electron-phonon coupling
enhancement and charge transfer. By visualizing and tuning the oxygen vacancies
at the interface, we find their very limited effect on the superconductivity,
which excludes interfacial oxygen vacancies as the primary source for charge
transfer between the substrate and FeSe films. Our findings have placed severe
constraints on any microscopic model for the high-$T_\textrm{c}$
superconductivity in FeSe-related heterostructures.
