The thesis presented herein utilizes a variety of methods to address the biomechanical function of vertebrate hind limb bones in both modern and fossil species. In an innovative application of beam theory, the maximum force a long bone is capable of withstanding before yield is calculated under a variety of simplified loading conditions for a phylogenetically diverse sample of modern birds and mammals. In doing so, new insights are gained into the combined role of limb bone geometry and load vector in achieving mass-invariant safety factors under static loading. In particular, the avian femur is found to scale with sufficient allometry such that no postural modification is required to maintain constant safety factors across several orders of magnitude in body mass. In a methodological study, two techniques for estimating stress (simple beam theory and finite element analysis, FEA) are compared across a sample of morphologically diverse long bones. The extent to which stress estimates derived from the two modelling techniques diverge is found to correlate to aspects of the underlying bone morphology such as shaft curvature and cross-sectional asymmetry, and important recommendations are made regarding the appropriate application of both methods to skeletal material. A novel 'convex hull' volumetric mass prediction technique for fossil birds is applied to two species of extinct moa (Dinornithiformes) from New Zealand. The resulting mass estimates are incorporated into a FEA study of the femora and tibiotarsi of modern ratites and moa. The 'stout southern' moa (Pachyornis australis) is confirmed as possessing extremely robust limbs, whilst the 'terrible robust' moa (Dinornis robustus) is found to possess equally, if not less, robust limb bones than those of modern ratites. The results are subsequently interpreted in the context of moa habitat range and shared ancestry. Finally the convex hull mass estimation technique is extended to modern primates, and the scaling of body mass with convex hull volume is compared across birds, primates and non-primate mammals. The allometric scaling of convex hull volume in birds and primates is considered in light of interspecific variation in muscle volume, body fat and integumentary structures, and is particularly relevant to those reconstructing the soft-tissue architecture of fossil species.