We present a detailed study of the Herbig-Haro objects HH 80/81, twin working surfaces in the highly collimated outflow driven by a luminous young star. High angular resolution emission line images obtained with the Hubble Space Telescope are used together with ground-based low- and high-dispersion spectroscopy, and proper motion measurements to provide a comprehensive picture of the structure and kinematics of these remarkable objects. The two principle knots HH 80A and HH 81A have emission lines with widths of 700 km s^-1 and 625 km s^-1 (FWZI), respectively—far broader than previously observed in any HH object—and also have large tangential velocities of about 350 km s^-1. In addition, they are both of exceptionally high excitation, having the strongest [O III] emission of any known HH object. Although the kinematics of these objects are broadly consistent with expectations from radiative bow shock models, the very high shock velocities implied, in excess of 600 km s^-1, mean that the postshock cooling distance at the apex is a few times greater than their size. Consequently, these bow shocks must have adiabatic tips and only become radiative in their wings. At the spatial resolution of our Hubble Space Telescope images the structure of HH 80/81 is seen to be far more complex than was thought on the basis of ground-based images. While HH 80A does bear some resemblance to a bow shock, HH 81A has an intricate filamentary structure. The HH 80/81 outflow is highly collimated from the source out to HH 81A and HH 80A, but then abruptly broadens into a network of faint shock excited streamers, terminating in a giant bow shock. The point at which the flow begins to diverge coincides with the apparent edge of the molecular cloud. We suggest that this morphology results because fast, very hot bullets like HH 80A and HH 81A violently expand as they escape from the cloud into its low pressure surroundings.