The Eynard-Orantin recursion is a universal mechanism of constructing a (re-modeld) B-model topological string theory on any given Riemann surface. The physics predictions tell us that its mirror symmetric partner, the A-model topological string theory corresponding to the starting B-model, should give us most everything about Gromov-Witten invariants of certain target spaces. Most recent speculations from physics include formulas for quantum knot invariants in terms of the starting Riemann surface (the A-polynomial). Surprisingly, from a purely algebro-goemmetric point of view, the Eynard-Orantin recursion appears often as the Laplace transform of the natural degeneration formulas of algebraic curves. This mysterious B-model formula is therefore a completely familiar equation to algebraic geometers. The only difference here is the Laplace transform, which plays the role of mirror symmetry. These talks are aimed at explaining this relation between algebraic geometry of curve degenerations and the re-modeled B-model topological string theory. We will present mathematically rigorous examples to illustrate the new developments and the exciting picture that is emerging now.