  We investigate the behaviour on approach to the initial singularity in
higher-order extensions of general relativity by finding exact cosmological
solutions for a wide class of models in which the Lagrangian is allowed to
depend nonlinearly upon the three possible linear and quadratic scalars built
from the Riemann tensor; $R$, $R_{ab}R^{ab}$ and $R_{abcd}R^{abcd}$. We present
new anisotropic vacuum solutions analagous to the Kasner solutions of general
relativity and extend previous results to a much wider range of fourth order
theories of gravity. We discuss the implications of these results for the
behaviour of the more general anisotropic Bianchi type VIII and IX cosmologies
as the initial singularity is approached. Furthermore, we also consider the
existence conditions for some other simple anisotropic Bianchi I vacuum
solutions in which the expansion in each direction is of exponential, rather
than power-law behaviour and their relevance for cosmic ``no-hair'' theorems.
