  We have reevaluated the hadronic contribution to the anomalous magnetic
moment of the muon (g-2) and to the running of the QED fine structure constant
alpha(s) at s=M_Z**2. We incorporated new data from hadronic tau decays,
recently published by the ALEPH Collaboration. In addition, compared to
previous analyses, we use more extensive e+e- annihilation data sets. The
integration over the total hadronic cross section is performed using
experimental data up to 40 GeV and results from perturbative QCD above 40 GeV.
The improvement from tau data concerns mainly the pion form factor, where the
uncertainty in the corresponding integral could be reduced by more than a
factor of two. We obtain for the lowest order hadronic vacuum polarization
graph a_mu(had) = (695.0 +/- 15.0) x 10^{-10} and delta(alpha(M_Z**2))(had) =
(280.9 +/- 6.3) x 10^{-4} using e+e- data only. The corresponding results for
combined e+e- and tau data are a_mu(had) = (701.1 +/- 9.4) x 10^{-10} and
delta(alpha(M_Z**2))(had) = (281.7 +/- 6.2) x 10^{-4}, where the latter is
calculated using the contribution from the five lightest quarks.

High Energy Physics - Phenomenology