Abstract: Topological Quantum Field Theory (TQFT), as formulated by Atiyah, has provided a general framework for understanding topological invariants of manifolds. The structure of TQFTs in dimension 1+1 (i.e. surfaces with boundaries) is completely understood by elementary means -- yet they can still yield surprising results. We present a famous example of a 1+1 dimensional TQFT that results in a beautiful old formula that counts covers of a genus g Riemann surface. Finally, we sketch how a deformation of this TQFT encodes the Gromov-Witten invariants of curves in Calabi-Yau 3-folds, and provides insight into the structure of Gromov-Witten invariants.