  The topology of Stein surfaces and contact 3-manifolds is studied by means of
handle decompositions. A simple characterization of homeomorphism types of
Stein surfaces is obtained --- they correspond to open handlebodies with all
handles of index lessthan or = 2. An uncountable collection of exotic R^4's is
shown to admit Stein structures. New invariants of contact 3-manifolds are
produced, including a complete (and computable) set of invariants for
determining the homotopy class of a 2-plane field on a 3-manifold. These
invariants are applicable to Seiberg-Witten theory. Several families of
oriented 3-manifolds are examined, namely the Seifert fibered spaces and all
surgeries on various links in S^3, and in each case it is seen that ``most''
members of the family are the oriented boundaries of Stein surfaces.
