  This is a survey paper focusing on the interplay between the curvature and
topology of a Riemannian manifold. The first part of the paper provides a
background discussion, aimed at non-experts, of Hopf's pinching problem and the
Sphere Theorem. In the second part, we sketch the proof of the Differentiable
Sphere Theorem, and discuss various related results. These results employ a
variety of methods, including geodesic and minimal surface techniques as well
as Hamilton's Ricci flow.
