Precision physics with QCD

The four-loop determination of the strong coupling from fully inclusive observables is reviewed. Special attention is given to the low-energy measurement extracted from the hadronic $\tau$ decay width. A recent exhaustive analysis of the ALEPH data, exploring several complementary methodologies with very different sensitivities to inverse power corrections and duality violations, confirms the strong suppression of non-perturbative contributions to $R_{\tau }$. It gives the value ${\alpha }_s(m_{\tau }^2)=0.328\pm 0.013$, which implies ${\alpha }_s(M_Z^2)=0.1197\pm 0.0015$. The excellent agreement with the direct measurement at the $Z$ peak, ${\alpha }_s(M_Z^2)=0.1196\pm 0.0030$, provides a beautiful test of asymptotic freedom. Together with the most recent lattice average from FLAG and the NNLO determinations from $e^{+}{e}^{-}$, PDFs and collider data quoted by the PDG, these two inclusive determinations imply a world average value ${\alpha }_s(M_Z^2)=0.1180\pm 0.0010$.