  We compute the shear viscosity of the unitary Fermi gas above the superfluid
transition temperature, using a diagrammatic technique that starts from the
exact Kubo formula. The formalism obeys a Ward identity associated with scale
invariance which guarantees that the bulk viscosity vanishes identically. For
the shear viscosity, vertex corrections and the associated Aslamazov-Larkin
contributions are shown to be crucial to reproduce the full Boltzmann equation
result in the high-temperature, low fugacity limit. The frequency dependent
shear viscosity $\eta(\omega)$ exhibits a Drude-like transport peak and a
power-law tail at large frequencies which is proportional to the Tan contact.
The weight in the transport peak is given by the equilibrium pressure, in
agreement with a sum rule due to Taylor and Randeria. Near the superfluid
transition the peak width is of the order of $0.5 T_F$, thus invalidating a
quasiparticle description. The ratio $\eta/s$ between the static shear
viscosity and the entropy density exhibits a minimum near the superfluid
transition temperature whose value is larger than the string theory bound
$\hbar/(4\pi k_B)$ by a factor of about seven.
