A hyperbolic surface is very flexible. There are many different
 hyperbolic structures (or equivalently conformal structures) on a compact
 surface. A compact hyperbolic manifold of dimension 4 or greater is very
 rigid. There are no deformations of any kind. Compact three dimensional
 hyperbolic manifolds lie somewhere in between. They are rigid if we
 restrict to smooth hyperbolic metrics. However, if we allow the metric to
 have a certain type of cone singularity deformations exist. In this talk
 we will give an outline of this deformation theory and explain how it can
 be used to to prove some classical conjectures about hyperbolic
 3-manifolds