Over the past decade, the discovery of a two-dimensional electron gas (2DEG) at the LaAlO₃/SrTiO₃ interface has resulted in the observation of remarkable properties, not present in conventional semiconductor heterostructures, and so become a focal point for device applications. Its counterpart, the two-dimensional hole gas (2DHG), is expected to complement the 2DEG. However, although the 2DEG has been widely observed, the 2DHG has proved elusive. In this talk, I will introduce a highly mobile 2DHG in epitaxially-grown SrTiO₃/LaAlO₃/SrTiO₃ heterostructures. Using electrical transport measurements and in-line electron holography, I provide direct evidence of a 2DHG that coexists with a 2DEG at complementary heterointerfaces in the same structure. The precise control of interfacial atomic structure and the removal of point defects are prerequisites to realizing 2DHG at oxide heterointerfaces. First-principles calculations, coherent Bragg rod analysis and depth-resolved cathodoluminescence spectroscopy consistently support our finding that to eliminate ionic point defects is key to realizing a 2DHG. The coexistence of a 2DEG and a 2DHG in a single oxide heterostructure provides a platform for the exciting physics of confined electron–hole systems and for developing applications.