  Simply-connected manifolds of positive sectional curvature $M$ are speculated
to have a rigid topological structure. In particular, they are conjectured to
be rationally elliptic, i.e., all but finitely many homotopy groups are
conjectured to be finite. In this article we combine positive curvature with
rational ellipticity to obtain several topological properties of the underlying
manifold. These results include a small upper bound on the Euler characteristic
and confirmations of famous conjectures by Hopf and Halperin under additional
torus symmetry. We prove several cases (including all known even-dimensional
examples of positively curved manifolds) of a conjecture by Wilhelm.
