  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 $\eta/s$
down by more than a factor of 3 at physically relevant couplings. The
perturbative expansion is problematic even at $T \simeq 100$ GeV. The largest
next-to-leading order correction to $\eta/s$ arises from modifications to the
qhat 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.
