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This talk is about a new family of geometric quantum link invariants that depend on both a link in S^3 and a flat sl_2 connection on its complement. When the connection is trivial they recover the Kashaev invariant (a certain evaluation of the colored Jones polynomial). More generally they can be understood as a quantization of the complex Chern-SImons invariant of the flat connection (aka complex volume), whose real and imaginary parts are the volume and Chern-Simons invariant of the hyperbolic structure determined by the connection. In this talk I will discuss the construction of these invariants using the representation theory of quantum sl_2 and how the classical complex Chern-Simons invariant arises naturally in this context, then sketch some proposed connections with quantum SL_2(C) Chern-Simons theory. Given time I will also discuss connections with the Volume Conjecture. This talk is based on joint work with Nicolai Reshetikhin.