  A theory of finite type invariants for arbitrary compact oriented 3-manifolds
is proposed, and illustrated through many examples arising from both classical
and quantum topology. The theory is seen to be highly non-trivial even for
manifolds with large first betti number, encompassing much of the complexity of
Ohtsuki's theory for homology spheres. (For example, it is seen that the
quantum SO(3) invariants, though not of finite type, are determined by finite
type invariants.) The algebraic structure of the set of all finite type
invariants is investigated, along with a combinatorial model for the theory in
terms of trivalent "Feynman diagrams".
