  We report on a major milestone in the construction of logarithmically
accurate final-state parton showers, achieving
next-to-next-to-leading-logarithmic (NNLL) accuracy for the wide class of
observables known as event shapes. The key to this advance lies in the
identification of the relation between critical NNLL analytic resummation
ingredients and their parton-shower counterparts. Our analytic discussion is
supplemented with numerical tests of the logarithmic accuracy of three shower
variants for more than a dozen distinct event-shape observables in $Z \to q
\bar q$ and Higgs $\to gg$ decays. The NNLL terms are phenomenologically
sizeable, as illustrated in comparisons to data.
