Oxide interfaces manifest many fascinating phenomena with synergetic correlations among multiple degrees of freedom, including the interplay of broken symmetry, lattice mismatch, charge transfer, spatial confinement. In particular, the interface superconductivity in oxide heterostructure has attracted extensive attention due to the rich underlying physical connotations. The interfaces not only provide alternative research platforms with respect to the bulk material counterpart for exploring new superconductors and investigating superconducting mechanisms, but also create new opportunities for applying superconductors to future electronic devices. In recent years, owing to the rapid development of heteroepitaxial techniques and accurate characterization methods, researchers have found quasi-two-dimensional interface superconductivity in various oxide heterostructures and revealed numerous novel quantum phenomena associated with interface superconductivity, which not only promotes the development of condensed matter physics, but also lays important foundation for the practical application of interface superconductivity. In this brief review, we mainly focus on the quasi-two-dimensional superconductivity at oxide interface. Taking the typical quasi-two-dimensional superconductivity at the LaAlO₃/SrTiO₃ interface and copper oxides such as La₂CuO₄/La_1.56Sr_0.44CuO₄ for example, we summarize and examine some novel physical phenomena with interface superconductivity in complex oxide heterostructures. Then we address the related problems that remain to be solved, and finally we prospect the possible future development of the interface superconductivity.