Given the challenges in experimental studies of uranium, the heaviest naturally occurring metal, we present first-principles calculation for the spin-dependent transport. Showing the largest atomic spin-orbit coupling we explore the ability of various crystal phases to maximize the charge-to-spin conversion using a fully relativistic Korringa-Kohn-Rostoker Green's function method. The transport theory is based on a semiclassical description where intrinsic and extrinsic, skew scattering, contributions can be separated easily. In addition to the various crystal phases we analyze the effect of substitutional impurities for gamma, hcp, as well as the alpha phase. We predict a very high, 10(4) (Omega cm)(-1), spin Hall conductivity for the metastable hcp-U phase, a giant value five times larger than for the conventional spin Hall material Pt. We estimated an efficiency of charge-to-spin current conversion of up to 30%. The spin diffusion length, a crucial parameter in any application, is predicted to be in the range from 3 to 6.5 nm, compatible with other charge-to-spin conversion materials. Relating our work to the sparse experimental results, our calculations suggest a gamma phase in the thin film rather than the experimentally expected alpha phase.