Part I: Positive scalar curvature.- The (moduli) space of all Riemannian metrics.- Clifford algebras and spin.- Dirac operators and index theorems.- Early results on the space of positive scalar curvature metrics.- Kreck-Stolz invariants.- Applications of Kreck-Stolz invariants.- The eta invariant and applications.- The case of dimensions 2 and 3.- The observer moduli space and applications.- Other topological structures.- Negative scalar and Ricci curvature.- Part II: Sectional curvature.- Moduli spaces of compact manifolds with positive or non-negative sectional curvature.- Moduli spaces of compact manifolds with negative and non-positive sectional curvature.- Moduli spaces of non-compact manifolds with non-negative sectional curvature.- Positive pinching and the Klingenberg-Sakai conjecture.