  This talk summarizes the recent developments in the evaluation of the leading
order hadronic contributions to the running of the QED fine structure constant
$\alpha(s)$, at $s=M_{\rm Z}^2$, and to the anomalous magnetic moment of the
muon $(g-2)_\mu$. The accuracy of the theoretical prediction of these
observables is limited by the uncertainties on the hadronic contributions.
Significant improvement has been achieved in a series of new analyses which is
presented historically in three steps: (I), use of $\tau$ spectral functions in
addition to $e^+e^-$ cross sections, (II), extended use of perturbative QCD and
(III), application of QCD sum rule techniques. The most precise values obtained
are: $\Delta\alpha_{\rm had}(M_{\rm Z}^2)$, $=(276.3\pm1.6)\times10^{-4}$,
yielding $\alpha^{-1}(M_{\rm Z}^2)=128.933\pm0.021$, and $a_\mu^{\rm
had}=(692.4\pm6.2)\times 10^{-10}$ with which one finds for the complete
Standard Model prediction $a_\mu^{\rm SM}=(11 659 159.6\pm6.7)\times10^{-10}$.
For the electron $(g-2)_e$, the hadronic contribution is $a_e^{\rm
had}=(187.5\pm1.8)\times 10^{-14}$.
