Abstract: After reviewing a few topological classification results (up to diffeomorphism, homeomorphism, or homotopy equivalence) for manifolds admitting metrics with positive sectional curvature and large isometry groups, I will discuss a couple of new results on weaker invariants (e.g., Betti numbers and Euler characteristics) of such manifolds. In one case, we obtain a sharp calculation of the Euler characteristic. This is joint work with Manuel Amann (KIT).