  In this paper we construct invariants of 3-manifolds "\`a la
Reshetikhin-Turaev" in the setting of non-semi-simple ribbon tensor categories.
We give concrete examples of such categories which lead to a family of
3-manifold invariants indexed by the integers. We prove this family of
invariants has several notable features, including: they can be computed via a
set of axioms, they distinguish homotopically equivalent manifolds that the
standard Reshetikhin-Turaev-Witten invariants do not, and they allow the
statement of a version of the Volume Conjecture and a proof of this conjecture
for an infinite class of links.
