These notes formed part of the ICTP summer school
on Geometry and Topology of 3-manifolds in June 2006. Assuming
only a minimal knowledge of hyperbolic geometry, the aim was
to provide a rapid introduction to the modern picture of Kleinian
groups. The subject has recently made dramatic progress with
spectacular proofs of the Density Conjecture, the Ending Lamination
Conjecture and the Tameness Conjecture. Between them,
these three new theorems make possible a complete geometric classification
of all hyperbolic 3-manifolds. The goal is to explain the
background needed to appreciate the statements and significance
of these remarkable results.
