Evaluation of $\alpha(M_{\rm Z}^2)$ and $(g-2)_\mu$

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 $a_e^{\rm had}=(187.5\pm1.8)\times 10^{-14}$.