A large impact by a comet or meteorite releases an enormous amount of energy, which evaporates, melts and fractures the surrounding rocks. Distinctive features of such impacts are 'shatter cones', deformed rocks characterized by hierarchical striated features. Although such features have been used for decades as unequivocal fingerprints of large-body impacts, the process by which shatter cones form has remained enigmatic. Here we show that the distinctive shatter-cone striations naturally result from nonlinear waves (front waves) that propagate along a fracture front. This explains the observed systematic increase of striation angles with the distance from the impact. Shatter-cone networks, typically spanning many scales, can be understood as hierarchical bifurcations of the fracture front, which is generated by the immense energy flux carried by the initial, impact-generated, shock waves. Our quantitative predictions based on this theory are supported by field measurements at the Kentland and Vredefort impact sites. These measurements indicate that shatter cones near to the impact site were formed by fractures propagating at nearly the Rayleigh wave speed of the host rocks, whereas the furthest shatter cones observed (about 40 km from the impact site) were formed by fronts moving more slowly. These results provide insight into impact dynamics as well as dissipative mechanisms in solids subjected to sudden, extremely intense fluxes of energy.