QCD shear viscosity at (almost) NLO

We compute the shear viscosity of QCD with matter, including almost all next-to-leading order corrections — that is, corrections suppressed by one power of g relative to leading order. We argue that the still missing terms are small. The next-to-leading order corrections are large and bring η/s down by more than a factor of 3 at physically relevant couplings. The perturbative expansion is problematic even at T ≃ 100 GeV. The largest next-to-leading order correction to η/s arises from modifications to the q^ parameter, which determines the rate of transverse momentum diffusion. We also explore quark number diffusion, and shear viscosity in pure-glue QCD and in QED.