  We study spaces and moduli spaces of Riemannian metrics with non-negative
Ricci or non-negative sectional curvature on closed and open manifolds. We
construct, in particular, the first classes of manifolds for which these moduli
spaces have non-trivial rational homotopy, homology and cohomology groups. We
also show that in every dimension at least seven (respectively, at least eight)
there exist closed (respectively, open) manifolds for which the space and
moduli space of Riemannian metrics with non-negative sectional curvature has
infinitely many path components. A completely analogous statement holds for
spaces and moduli spaces of non-negative Ricci curvature metrics.
