The local Gromov-Witten theory of curves is solved by localization and degeneration methods. Localization is used for the exact evaluation of basic integrals in the local Gromov-Witten theory of P1. A TQFT formalism is defined via degeneration to capture higher genus curves. Together, the results provide a compete and effective solution.

The local Gromov-Witten theory of curves is equivalent to the local Donaldson-Thomas theory of curves, the quantum cohomology of the Hilbert scheme points in the plane, and the orbifold quantum cohomology the symmetric product of the plane. The results of the paper provide the local Gromov-Witten calculations required for the proofs of these equivalences.