  We calculate the far-from-equilibrium dynamics and thermalization both for
the quantum and the classical O(N)--model. The early and late-time behavior can
be described from the 2PI--loop expansion for weak couplings or the
nonperturbative 2PI--1/N expansion of the effective action beyond leading
order. A comparison with exact simulations in 1+1 dimensions in the classical
limit shows that the 2PI--1/N expansion at next-to-leading order gives
quantitatively precise results already for moderate values of N. We derive a
criterion for the validity of the classical approximation and verify it by
comparing far-from-equilibrium quantum and classical dynamics. At late times
one observes the expected deviations due to the difference between classical
and quantum thermal equilibrium.
