  In this paper we give, for the first time, a complete description of the
late-time evolution of non-tilted spatially homogeneous cosmologies of Bianchi
type VIII. The source is assumed to be a perfect fluid with equation of state
$p = (\gamma - 1)\mu$, where $\gamma$ is a constant which satisfies $1 \leq
\gamma \leq 2$. Using the orthonormal frame formalism and Hubble-normalized
variables, we rigorously establish the limiting behaviour of the models at late
times, and give asymptotic expansions for the key physical variables. The main
result is that asymptotic self-similarity breaking occurs, and is accompanied
by the phenomenon of `Weyl curvature dominance', characterized by the
divergence of the Hubble-normalized Weyl curvature at late times.
