  The volume conjecture states that for a hyperbolic knot K in the three-sphere
S^3 the asymptotic growth of the colored Jones polynomial of K is governed by
the hyperbolic volume of the knot complement S^3\K. The conjecture relates two
topological invariants, one combinatorial and one geometric, in a very
nonobvious, nontrivial manner. The goal of the present lectures is to review
the original statement of the volume conjecture and its recent extensions and
generalizations, and to show how, in the most general context, the conjecture
can be understood in terms of topological quantum field theory. In particular,
we consider: a) generalization of the volume conjecture to families of
incomplete hyperbolic metrics; b) generalization that involves not only the
leading (volume) term, but the entire asymptotic expansion in 1/N; c)
generalization to quantum group invariants for groups of higher rank; and d)
generalization to arbitrary links in arbitrary three-manifolds.
