  The frequency-dependent longitudinal and Hall conductivities ---
$\sigma_{xx}$ and $\sigma_{xy}$ --- are dimensionless functions of $\omega/T$
in 2+1 dimensional CFTs at nonzero temperature. These functions characterize
the spectrum of charged excitations of the theory and are basic experimental
observables. We compute these conductivities for large $N$ Chern-Simons theory
with fermion matter. The computation is exact in the 't Hooft coupling
$\lambda$ at $N = \infty$. We describe various physical features of the
conductivity, including an explicit relation between the weight of the delta
function at $\omega = 0$ in $\sigma_{xx}$ and the existence of infinitely many
higher spin conserved currents in the theory. We also compute the
conductivities perturbatively in Chern-Simons theory with scalar matter and
show that the resulting functions of $\omega/T$ agree with the strong coupling
fermionic result. This provides a new test of the conjectured 3d bosonization
duality. In matching the Hall conductivities we resolve an outstanding puzzle
by carefully treating an extra anomaly that arises in the regularization scheme
used.
