  We prove several basic ring-theoretic results about tautological rings of
manifolds W, that is, the rings of generalised Miller--Morita--Mumford classes
for fibre bundles with fibre W. Firstly we provide conditions on the rational
cohomology of W which ensure that its tautological ring is finitely-generated,
and we show that these conditions cannot be completely relaxed by giving an
example of a tautological ring which fails to be finitely-generated in quite a
strong sense. Secondly, we provide conditions on torus actions on W which
ensure that the rank of the torus gives a lower bound for the Krull dimension
of the tautological ring of W. Lastly, we give extensive computations in the
tautological rings of CP^2 and S^2 x S^2.
