Since the dramatic enhancement of the superconducting transition temperature (T c) was reported in a one-unit-cell FeSe film grown on a SrTiO3 substrate (1-UC FeSe/STO) by molecular beam epitaxy (MBE), related research on this system has become a new frontier in condensed matter physics. In this paper, we present a brief review on this rapidly developing field, mainly focusing on the superconducting properties of 1-UC FeSe/STO. Experimental evidence for high-temperature superconductivity in 1-UC FeSe/STO, including direct evidence revealed by transport and diamagnetic measurements, as well as other evidence from scanning tunneling microscopy (STM) and angle-resolved photoemission spectroscopy (ARPES), are overviewed. The potential mechanisms of the enhanced superconductivity are also discussed. There are accumulating arguments to suggest that the strengthened Cooper pairing in 1-UC FeSe/STO originates from the interface effects, specifically the charge transfer and coupling to phonon modes in the TiO2 plane. The study of superconductivity in 1-UC FeSe/STO not only sheds new light on the mechanism of high-temperature superconductors with layered structures, but also provides an insight into the exploration of new superconductors by interface engineering.