alpha_s$(2019): Precision measurements of the QCD coupling

This document collects a written summary of all contributions presented at the workshop "$\alpha_s$(2019): Precision measurements of the strong coupling" held at ECT* (Trento) in Feb. 11--15, 2019. The workshop explored in depth the latest developments on the determination of the QCD coupling $\alpha_s$ from the key categories where high precision measurements are available: (i) lattice QCD, (ii) hadronic $\tau$ decays, (iii) deep-inelastic scattering and parton distribution functions, (iv) event shapes, jet cross sections, and other hadronic final-states in $e^+e^-$ collisions, (v) Z boson and W boson hadronic decays, and (vi) hadronic final states in p-p collisions. The status of the current theoretical and experimental uncertainties associated to each extraction method, and future perspectives were thoroughly reviewed. Novel $\alpha_s$ determination approaches were discussed, as well as the combination method used to obtain a world-average value of the QCD coupling at the Z mass pole.