Metallic uranium-zirconium alloys are of interest for a variety of fast reactor designs, and there is substantial experience with the behavior of metallic fuels. Yet, there remain a number of questions regarding the mechanisms controlling fission-gas-driven swelling in these alloys. Here we present results of ab initio calculations of the diffusion behavior of interstitial and vacancy point defects in α U-Zr alloys. The formation energy and migration barrier of vacancy and interstitial defects, and the influence of Zr on these values, is obtained and compared with experimental results. Our results confirm that self-diffusion in pure α U is via a simple vacancy mechanism, and shows anisotropic character. The calculated values of activation energy are consistent with the experimental results in the literature. For interstitial diffusion, the kick-out mechanism was found to have the smallest energy barrier. The calculations of point defects, and later Xe, in U-Zr alloys will provide a foundation for computational modeling of fission gas bubble nucleation and growth.