  We revisit the determination of alpha_s(m_tau) using a fit to inclusive tau
hadronic spectral moments in light of (1) the recent calculation of the
fourth-order perturbative coefficient K_4 in the expansion of the Adler
function, (2) new precision measurements from BABAR of e+e- annihilation cross
sections, which decrease the uncertainty in the separation of vector and
axial-vector spectral functions, and (3) improved results from BABAR and Belle
on tau branching fractions involving kaons. We estimate that the fourth-order
perturbative prediction reduces the theoretical uncertainty, introduced by the
truncation of the series, by 20% with respect to earlier determinations. We
discuss to some detail the perturbative prediction and show that the effect of
the incomplete knowledge of the series is reduced by using the so-called
contour-improved calculation, as opposed to fixed-order perturbation theory
which manifests convergence problems. The corresponding theoretical
uncertainties are studied at the tau and Z mass scales. Nonperturbative
contributions extracted from the most inclusive fit are small, in agreement
with earlier determinations. Systematic effects from quark-hadron duality
violation are estimated with simple models and found to be within the quoted
systematic errors. The fit gives alpha_s(m_tau) = 0.344 +- 0.005 +- 0.007,
where the first error is experimental and the second theoretical. After
evolution to M_Z we obtain alpha_s(M_Z) = 0.1212 +- 0.0005 +- 0.0008 +- 0.0005,
where the errors are respectively experimental, theoretical and due to the
evolution. The result is in agreement with the corresponding NNNLO value
derived from essentially the Z width in the global electroweak fit. The
alpha_s(M_Z) determination from tau decays is the most precise one to date.
