We use invariants related to eta invariants of Dirac operators to distinguish path components of moduli spaces of Riemannian metrics with positive and nonnegative Ricci and sectional curvature. In 7 dimensions, we calculate the Kreck-Stolz s invariant for metrics on spin total spaces of Sn bundles with nonnegative sectional curvature. We then apply it to show that the moduli spaces of metrics with nonnegative sectional curvature on certain 7-manifolds have infinitely many path components. These include certain positively curved Eschenburg and Aloff-Wallach spaces. We next use the eta invariant of spinc Dirac operators to distinguish connected components of moduli spaces of Riemannian metrics with positive Ricci curvature. We find infinitely many non-diffeomorphic five dimensional manifolds for which these moduli spaces each have infinitely many components. The manifolds are total spaces of principal S^1 bundles over #^aCP2#^bCP2 and the metrics are lifted from Ricci positive metrics on the base.