Intersubband transitions in n-doped multi-quantum-well semiconductor heterostructures make it possible to engineer one of the largest known nonlinear optical responses in condensed matter systems--but this nonlinear response is limited to light with electric field polarized normal to the semiconductor layers. In a different context, plasmonic metasurfaces (thin conductor-dielectric composite materials) have been proposed as a way of strongly enhancing light-matter interaction and realizing ultrathin planarized devices with exotic wave properties. Here we propose and experimentally realize metasurfaces with a record-high nonlinear response based on the coupling of electromagnetic modes in plasmonic metasurfaces with quantum-engineered electronic intersubband transitions in semiconductor heterostructures. We show that it is possible to engineer almost any element of the nonlinear susceptibility tensor of these structures, and we experimentally verify this concept by realizing a 400-nm-thick metasurface with nonlinear susceptibility of greater than 5 × 10(4) picometres per volt for second harmonic generation at a wavelength of about 8 micrometres under normal incidence. This susceptibility is many orders of magnitude larger than any second-order nonlinear response in optical metasurfaces measured so far. The proposed structures can act as ultrathin highly nonlinear optical elements that enable efficient frequency mixing with relaxed phase-matching conditions, ideal for realizing broadband frequency up- and down-conversions, phase conjugation and all-optical control and tunability over a surface.