In a series of papers, Gopakumar, Ooguri and Vafa proposed the existence of a duality between two quantum mechanical models: a topological gauge theory - SU(N) Chern-Simons theory on the three-sphere - on one hand, and a topological string theory - the topological A-model on the so-called "resolved conifold" - on the other. From a physical point of view, the duality provides a concrete instance of the gauge/string correspondence, and one where exact computations can be performed in detail. Mathematically, this connection puts forward a triangle of striking, conjectural relations between quantum invariants (Reshetikhin-Turaev-Witten) of knots and 3-manifolds, curve-counting invariants (Gromov-Witten/Donaldson-Thomas) of some local Calabi-Yau 3-folds, and the Eynard-Orantin topological recursion for a specific class of spectral curves. I will survey the status of the conjecture and discuss extensions of (and obstructions to) this circle of ideas to the case of Chern-Simons theory on spherical 3-manifolds.