  The dynamics of cosmological models with isotropic matter sources (perfect
fluids) is extensively studied in the literature; in comparison, the dynamics
of cosmological models with anisotropic matter sources is not. In this paper we
consider spatially homogeneous locally rotationally symmetric solutions of the
Einstein equations with a large class of anisotropic matter models including
collisionless matter (Vlasov), elastic matter, and magnetic fields. The
dynamics of models of Bianchi types I, II, and IX are completely described; the
two most striking results are the following: (i) There exist matter models,
compatible with the standard energy conditions, such that solutions of Bianchi
type IX (closed cosmologies) need not necessarily recollapse; there is an open
set of forever expanding solutions. (ii) Generic type IX solutions associated
with a matter model like Vlasov matter exhibit oscillatory behavior toward the
initial singularity. This behavior differs significantly from that of
vacuum/perfect fluid cosmologies; hence "matter matters". Finally, we indicate
that our methods can probably be extended to treat a number of open problems,
in particular, the dynamics of Bianchi type VIII and Kantowski-Sachs solutions.
