  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 P^1. 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 of C^2, and the orbifold quantum cohomology the symmetric product of
C^2. The results of the paper provide the local Gromov-Witten calculations
required for the proofs of these equivalences.
