Searching for superconducting materials with high transition temperature (TC) is one of the most exciting and challenging fields in physics and materials science. Although superconductivity has been discovered for more than 100 years, the copper oxides are so far the only materials with TC above 77 K, the liquid nitrogen boiling point. Here we report an interface engineering method for dramatically raising the TC of superconducting films. We find that one unit-cell (UC) thick films of FeSe grown on SrTiO3 (STO) substrates by molecular beam epitaxy (MBE) show signatures of superconducting transition above 50 K by transport measurement. A superconducting gap as large as 20 meV of the 1 UC films observed by scanning tunneling microcopy (STM) suggests that the superconductivity could occur above 77 K. The occurrence of superconductivity is further supported by the presence of superconducting vortices under magnetic field. Our work not only demonstrates a powerful way for finding new superconductors and for raising TC, but also provides a well-defined platform for systematic study of the mechanism of unconventional superconductivity by using different superconducting materials and substrates.