  A fundamental question in Riemannian geometry is to find canonical metrics on
a given smooth manifold. In the 1980s, R. Hamilton proposed an approach to this
question based on parabolic partial differential equations. The goal is to
start from a given initial metric and deform it to a canonical metric by means
of an evolution equation. There are various natural evolution equations for
Riemannian metrics, including the Ricci flow and the conformal Yamabe flow. In
this survey, we discuss the global behavior of the solutions to these
equations. In particular, we describe how these techniques can be used to prove
the Differentiable Sphere Theorem.
  This article is based on the Takagi Lectures delivered by the author at the
Research Institute for Mathematical Sciences, Kyoto University, on June 4,
2011.
