  We have computed the hadronic light-by-light (LbL) contribution to the muon
anomalous magnetic moment $a_\mu$ in the frame of Chiral Perturbation Theory
with the inclusion of the lightest resonance multiplets as dynamical fields
(R$\chi$T). It is essential to give a more accurate prediction of this hadronic
contribution due to the future projects of J-Parc and FNAL on reducing the
uncertainty in this observable. We, therefore, computed the pseudoscalar
transition form factor and proposed the measurement of the
$e^+e^-\to\mu^+\mu^-\pi^0$ cross section and dimuon invariant mass spectrum to
determine more accurately its parameters. Then, we evaluated the pion exchange
contribution to $a_\mu$, obtaining $(6.66\pm0.21)\cdot10^{-10}$. By comparing
the pion exchange contribution and the pion-pole approximation to the
corresponding transition form factor ($\pi$TFF) we recalled that the latter
underestimates the complete $\pi$TFF by (15-20)\%. Then, we obtained the
$\eta^({}'^)$ TFF, obtaining a total contribution of the lightest pseudoscalar
exchanges of $(10.47\pm0.54)\cdot10^{-10}$, in agreement with previous results
and with smaller error.
