  In this paper, we present foundational material towards the development of a
rigorous enumerative theory of stable maps with Lagrangian boundary conditions,
ie stable maps from bordered Riemann surfaces to a symplectic manifold, such
that the boundary maps to a Lagrangian submanifold. Our main application is to
a situation where our proposed theory leads to a well-defined algebro-geometric
computation very similar to well-known localization techniques in Gromov-Witten
theory. In particular, our computation of the invariants for multiple covers of
a generic disc bounding a special Lagrangian submanifold in a Calabi-Yau
threefold agrees completely with the original predictions of Ooguri and Vafa
based on string duality. Our proposed invariants depend more generally on a
discrete parameter which came to light in the work of Aganagic, Klemm, and Vafa
which was also based on duality, and our more general calculations agree with
theirs up to sign.
