We report a first-principles, periodic supercell analysis of oxygen adsorption, diffusion, and dissociation at the kinked Pt(321) surface. Binding energies and binding site preferences of isolated oxygen atoms and molecules have been determined, and we show that both atomic and molecular oxygen prefer binding in bridge sites involving coordinatively unsaturated kink Pt atoms. Binding energies of atomic and molecular oxygen in different sites correlate well with the average metallic Pt coordination number of Pt atoms forming each site, although differences exist between adsorbates in symmetrically similar sites due to the inherent chirality of the surface. Atomic O in the strongest binding bridge sites experiences relatively small energy barriers for diffusion to neighboring sites compared to O on Pt(111). However, due to the structure of the surface, O diffusion is only rapid between different sites around the kink Pt atom, whereas the effective long-range tracer diffusion, as determined from a simple course-grain model, is shown to be anisotropic and slower than on the Pt(111) surface. Four dissociation pathways for O(2) at low coverage are also reported and found to be in agreement with experimental observations of facile dissociation, even at low temperature.