One fundamental problem is to construct equivalences between
  manifolds; another is to construct invariants.  Of course these
  complementary problems should be related. This talk will be general
  discussion of some developments in these directions. We recall the
  special property of the curvature tensor of a Riemannian 3-manifold
  which underpins the Geometrisation Theorem. Then we discuss
  Riemannian and other structures in higher dimensions.  We give an
  account of work of Gromov, Taubes, and others, using holomorphic
  spheres in symplectic 4-manifolds, and of results of Friedl, and
  Vidussi, and others, on fibred 3-manifolds and symplectic
  structures.