I will report on results and questions about spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds and present recent joint work with Michael Wiemeler. We construct the first classes of manifolds for which these 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 moduli space of Riemannian metrics with non-negative sectional curvature has infinitely many path components.