The insulating state is one of the most basic electronic phases in condensed matter. This state is characterised by an energy gap for electronic excitations that makes an insulator electrically inert at low energy. However, for complex oxides, the very concept of an insulator must be re-examined. Complex oxides behave differently from conventional insulators such as SiO2, on which the entire semiconductor industry is based, because of the presence of multiple defect levels within their band gap. As the semiconductor industry is moving to such oxides for high-dielectric (high-k) materials, we need to truly understand the insulating properties of these oxides under various electric field excitations. Here we report a new class of material called nonlinear insulators that exhibits a reversible electric-field-induced metal-insulator transition. We demonstrate this behaviour for an insulating LaAlO3 thin film in a metal/LaAlO3/Nb-SrTiO3 heterostructure. Reproducible transitions were observed between a low-resistance metallic state and a high-resistance non-metallic state when applying suitable voltages. Our experimental results exclude the possibility that diffusion of the metal electrodes or oxygen vacancies into the LaAlO3 layer is occurring. Instead, the phenomenon is attributed to the formation of a quasi-conduction band (QCB) in the defect states of LaAlO3 that forms a continuum state with the conduction band of the Nb-SrTiO3. Once this continuum (metallic) state is formed, the state remains stable even when the voltage bias is turned off. An opposing voltage is required to deplete the charges from the defect states. Our ability to manipulate and control these defect states and, thus, the nonlinear insulating properties of complex oxides will open up a new path to develop novel devices.