  We study the evolution of the Weyl curvature invariant in all spatially
homogeneous universe models containing a non-tilted gamma-law perfect fluid. We
investigate all the Bianchi and Thurston type universe models and calculate the
asymptotic evolution of Weyl curvature invariant for generic solutions to the
Einstein field equations. The influence of compact topology on Bianchi types
with hyperbolic space sections is also considered. Special emphasis is placed
on the late-time behaviour where several interesting properties of the Weyl
curvature invariant occur. The late-time behaviour is classified into five
distinctive categories. It is found that for a large class of models, the
generic late-time behaviour the Weyl curvature invariant is to dominate the
Ricci invariant at late times. This behaviour occurs in universe models which
have future attractors that are plane-wave spacetimes, for which all scalar
curvature invariants vanish. The overall behaviour of the Weyl curvature
invariant is discussed in relation to the proposal that some function of the
Weyl tensor or its invariants should play the role of a gravitational 'entropy'
for cosmological evolution. In particular, it is found that for all
ever-expanding models the measure of gravitational entropy proposed by Gron and
Hervik increases at late times.
