Using an inverse dynamics biomechanical analysis that was previously validated for extant bipeds, I calculated the minimum amount of actively contracting hindlimb extensor muscle that would have been needed for rapid bipedal running in several extinct dinosaur taxa. I analyzed models of nine theropod dinosaurs (including birds) covering over five orders of magnitude in size. My results uphold previous findings that large theropods such as Tyrannosaurus could not run very quickly, whereas smaller theropods (including some extinct birds) were adept runners. Furthermore, my results strengthen the contention that many nonavian theropods, especially larger individuals, used fairly upright limb orientations, which would have reduced required muscular force, and hence muscle mass. Additional sensitivity analysis of muscle fascicle lengths, moment arms, and limb orientation supports these conclusions and points out directions for future research on the musculoskeletal limits on running ability. Although ankle extensor muscle support is shown to have been important for all taxa, the ability of hip extensor muscles to support the body appears to be a crucial limit for running capacity in larger taxa. I discuss what speeds were possible for different theropod dinosaurs, and how running ability evolved in an inverse relationship to body size in archosaurs. J. Morphol. 262:441–461, 2004. © 2004 Wiley-Liss, Inc.