I will report on recently discovered relations between closed Gromov-Witten theory of a Fano variety and open Gromov-Witten theory of Lagrangian submanifolds contained in it. The focus will be on the result saying that the quantum period of a Fano variety equals the classical period of the Landau-Ginzburg potential of any monotone Lagrangian torus sitting inside. This has applications "in both directions", including the classification of potentials of tori in CP2, and a proof of the quantum Lefschetz hyperplane theorem in the symplectic category.