  Contact structures on 3-manifolds are analyzed by decomposing the manifold
along convex surfaces. Background results of Giroux, Eliashberg, Colin, and
Honda are discussed with an emphasis on examples. Convex decompositions are
then used to give a new proof of the Gabai-Eliashberg-Thurston Theorem on the
existence of universally tight contact structures and also to study the contact
topology of a space in the presence or absence of tori. Classification of tight
contact structures on fibred manifolds and related open questions are also
discussed. This paper is based on a series of talks given at the Tokyo
Institute of Technology from Jun 3-7, 2002.
