  We compute the non-zero temperature conductivity of conserved flavor currents
in conformal field theories (CFTs) in 2+1 spacetime dimensions. At frequencies
much greater than the temperature, $\hbar\omega>> k_B T$, the $\omega$
dependence can be computed from the operator product expansion (OPE) between
the currents and operators which acquire a non-zero expectation value at T > 0.
Such results are found to be in excellent agreement with quantum Monte Carlo
studies of the O(2) Wilson-Fisher CFT. Results for the conductivity and other
observables are also obtained in vector 1/N expansions. We match these large
$\omega$ results to the corresponding correlators of holographic
representations of the CFT: the holographic approach then allows us to
extrapolate to small $\hbar \omega/(k_B T)$. Other holographic studies
implicitly only used the OPE between the currents and the energy-momentum
tensor, and this yields the correct leading large $\omega$ behavior for a large
class of CFTs. However, for the Wilson-Fisher CFT a relevant "thermal" operator
must also be considered, and then consistency with the Monte Carlo results is
obtained without a previously needed ad hoc rescaling of the T value. We also
establish sum rules obeyed by the conductivity of a wide class of CFTs.
