An intense laser field can remove an electron from an atom or molecule and pull the electron into a large-amplitude oscillation in which it repeatedly collides with the charged core it left behind(1-4). Such recollisions result in the emission of very energetic photons by means of high-order-harmonic generation, which has been observed in atomic and molecular gases(5-7) as well as in a bulk crystal(8). An exciton is an atom-like excitation of a solid in which an electron that is excited from the valence band is bound by the Coulomb interaction to the hole it left behind(9,10). It has been predicted that recollisions between electrons and holes in excitons will result in a new phenomenon: high-order-sideband generation(11,12). In this process, excitons are created by a weak near-infrared laser of frequency f(NIR). An intense laser field at a much lower frequency, f(THz), then removes the electron from the exciton and causes it to recollide with the resulting hole. New emission is predicted to occur as sidebands of frequency f(NIR) + 12nf(THz), wheren is an integer that can be much greater than one. Here we report the observation of high-order-sideband generation in semiconductor quantum wells. Sidebands are observed up to eighteenth order (+18f(THz), or n 59). The intensity of the high-order sidebands decays only weakly with increasing sideband order, confirming the non-perturbative nature of the effect. Sidebands are strongest for linearly polarized terahertz radiation and vanish when the terahertz radiation is circularly polarized. Beyond their fundamental scientific significance, our results suggest a new mechanism for the ultrafast modulation of light, which has potential applications in terabit-rate optical communications.