This work is a step towards a non-perturbative continuum definition of quantum field theory (QFT), beginning with asymptotically free two dimensional non-linear sigma-models, using recent ideas from mathematics and QFT. The ideas from mathematics are resurgence theory, the trans-series framework, and Borel-Ecalle resummation. The ideas from QFT use continuity on R^1 x S^1_L, i.e, the absence of any phase transition as N \to infinity, or rapid-crossovers for finite-N, and the small-L weak coupling limit to render the semi-classical sector well-defined and calculable. We classify semi-classical configurations with actions 1/N (kink-instantons), 2/N (bions and bi-kinks), in units where the 2d instanton action is normalized to one. Perturbation theory possesses the IR-renormalon ambiguity that arises due to non-Borel summability of the large-orders perturbation series (of Gevrey-1 type), for which a microscopic cancellation mechanism was unknown. This divergence must be present because the corresponding expansion is on a singular Stokes ray in the complexified coupling constant plane, and the sum exhibits the Stokes phenomenon crossing the ray. We show that there is also a non-perturbative ambiguity inherent to certain neutral topological molecules (neutral bions and bion-anti-bions) in the semiclassical expansion. We find a set of "confluence equations" that encode the exact cancellation of the two different type of ambiguities. We show that a new notion of "graded resurgence triangle" is necessary to capture the path integral approach to resurgence, and that graded resurgence underlies a potentially rigorous definition of general QFTs. The mass gap and the Theta angle dependence of vacuum energy are calculated from first principles, and are in accord with large-N and lattice results.