I will discuss spaces and moduli spaces of Riemannian metrics with
lower curvature bounds and their connectedness properties
and, in particular, the following recent
result which is joint work with Anand Dessai and Stephan Klaus:
In each dimension $4n+3$, $n\geq 1$, there exist closed simply connected manifolds
for which the moduli space of Riemannian metrics with nonnegative sectional curvature
has infinitely many path components. Spaces with these properties were known before only in dimension seven,
and our result does also hold for moduli spaces of Riemannian metrics with positive Ricci curvature.