  Starting from a factorization theorem in Soft-Collinear Effective Theory, the
thrust distribution in e+e- collisions is calculated including resummation of
the next-to-next-to-next-to leading logarithms. This is a significant
improvement over previous calculations which were only valid to next-to-leading
logarithmic order. The fixed-order expansion of the resummed result approaches
the exact fixed-order distribution towards the kinematic endpoint. This close
agreement provides a verification of both the effective field theory expression
and recently completed next-to-next-to-leading fixed order event shapes. The
resummed distribution is then matched to fixed order, resulting in a
distribution valid over a large range of thrust. A fit to ALEPH and OPAL data
from LEP 1 and LEP 2 produces alpha_s(m_Z)= 0.1172 +/- 0.0010 +/- 0.0008
+/-0.0012 +/- 0.0012, where the uncertainties are respectively statistical,
systematic, hadronic, and perturbative. This is one of the world's most precise
extractions of alpha_s to date.
