Conventional three-dimensional (3D) structure determination methods require either multiple measurements at different sample orientations or a collection of serial sections through a sample. Here we report the experimental demonstration of single-shot 3D structure determination of an object; in this case, individual gold nanocrystals at ~5.5 nm resolution using ~10 fs X-ray free-electron laser pulses. Coherent diffraction patterns are collected from high-index-faceted nanocrystals, each struck by an X-ray free-electron laser pulse. Taking advantage of the symmetry of the nanocrystal and the curvature of the Ewald sphere, we reconstruct the 3D structure of each nanocrystal from a single-shot diffraction pattern. By averaging a sufficient number of identical nanocrystals, this method may be used to determine the 3D structure of nanocrystals at atomic resolution. As symmetry exists in many virus particles, this method may also be applied to 3D structure studies of such particles at nanometer resolution on femtosecond time scales.