A resurgence function is a germ of an analytic function with an endless analytic continuation in the complex plane minus a discrete set of points. Resurgent functions were introduced by J. Ecalle who proved that general solutions of differential/difference equations (linear or not) are resurgent. 3-dimensional quantum field theory associates two power series to each knotted object (ie to a knot or a 3-manifold). We will formulate a resurgence conjecture for these two power series, and a precise analytic relation among them. Our resurgence conjecture implies in particular the volume conjecture, and a conjecture of Witten. Moreover, the set of singularities of our resurgent series have geometric meaning (they are values of the complex Chern-Simons action on flat connections), as well as algebraic meaning (they are given by the dilogarithm function at algebraic numbers).