Abstract: I will discuss aspects of Chern-Simons theory with complex gauge group, and the rich interaction between mathematics and physics that has fueled its development in recent years. Like its close cousin with compact gauge group, complex Chern-Simons is a topological field theory in three dimensions. Its partition functions on a large class of three-manifolds can be defined by convergent, finite-dimensional integrals, providing a new class of topological invariants with remarkable properties. Their perturbative expansions are arithmetic and (conjecturally) display modular properties. The invariants are closely related to cluster algebras and to Teichmuller theory. The physics of complex Chern-Simons theory also suggests that the invariants admit a categorification (analogous to Khovanov homology), which has been found in some simple cases.