  The Blaschke conjecture claims that every compact Riemannian manifold whose
injectivity radius equals its diameter is, up to constant rescaling, a compact
rank one symmetric space. We summarize the intuition behind this problem, the
proof that such manifolds have the cohomology of compact rank one symmetric
spaces, and the proof of the conjecture for homology spheres and homology real
projective spaces. We also summarize what is known on the diffeomorphism,
homeomorphism and homotopy types of such manifolds.
