Moduli spaces of metrics with nonnegative sectional or Ricci curvature

I will present results obtained within the project about the topology of the moduli space of metrics satisfying a lower curvature bound. The focus will be on manifolds that are known to admit metrics of nonnegative/positive sectional or Ricci curvature. I will describe different classes of manifolds where the respective moduli space has non-trivial topology, e.g., has infinitely many connected components or non-trivial higher invariants. I will also try to explain some of the key ideas used in the proofs, as well as the limitations one faces when working with these curvature bounds.