The temperature- and pressure-dependent kinetics of the reaction between propargyl and molecular oxygen have been studied with a combination of electronic structure theory, transition state theory, and the time-dependent master equation. The stationary points on the potential energy surface were located with B3LYP density functional theory. Approximate QCISD(T,Full)/6-311++G(3df,2pd) energies were obtained at these stationary points. At low temperatures the reaction is dominated by addition to the CH2 side of the propargyl radical followed by stabilization. However, addition to the CH side, which is followed by one of various possible internal rearrangements, becomes the dominant process at higher temperatures. These internal rearrangements involve a splitting of the O2 bond via the formation of 3-, 4- or 5-membered rings, with the apparent products being CH2CO + HCO. Rearrangement via the 3-membered ring is found to dominate the kinetics. Rearrangement from the CH2 addition product, via a 4-membered ring, would yield H2CO + HCCO, but the barrier to this rearrangement is too high to be kinetically significant. Other possible products require H transfers and, as a result, appear to be kinetically irrelevant. Modest variations in the energetics of a few key stationary points (most notably the entrance barrier heights) yield kinetic results that are in good agreement with the experimental results of Slagle and Gutman (I. R. Slagle and D. Gutman, Proc. Combust. Inst., 1986, 21, 875) and of Atkinson and Hudgens (D. B. Atkinson and J. W. Hudgens, J. Phys. Chem. A, 1999, 103, 4242).