Shatter cones are rock discontinuities known only from sites of extraterrestrial impacts, and they are assumed to be formed by impact-induced shock waves. Here we characterize the structure of shatter cones by field and microanalyses and explain their formation by dynamic fracture mechanics. Our analyses reveal that shatter cones always occur as multilevel, three-dimensional networks, 0.01--100 m in size, with hierarchal branched fractures. A typical, individual shatter cone is a curved, oblate branch that bifurcates from its parent fracture (e.g., a larger shatter cone) and expands to form a spoon-like surface. The unique shatter cone striations are arranged in V-shaped pairs whose enclosed angle is constant for a given sample. We propose that shatter cones are the natural consequence of tensile rock fracturing at extreme velocities. First, the structure of shatter cone networks is strikingly similar to the structure of branched networks of experimental dynamic fractures that propagate at high velocities (velocities that approach the Rayleigh wave speed, VR). Second, fracture front waves, generated experimentally by the interaction of a rapidly moving tensile fractures and material inclusions, create tracks on the fracture surface that correspond to the V-shaped striations of shatter cones. Third, applying the front wave concept to our field measurements (Vredefort impact, South Africa) shows that the shatter cones propagated at velocities of 0.98--0.90VR, with a systematic velocity decrease from the impact center. These extreme asymptotic velocities require the intense energy flux of impacts. Our model explains all of the structural features of shatter cones (curved surfaces, cone directivity, unique striations, hierarchic, multilevel structure) and their exclusive occurrence at impact sites.