  Calculations of the number of curves on a Calabi-Yau manifold via an
instanton expansion do not always agree with what one would expect naively. It
is explained how to account for continuous families of instantons via
deformation theory and excess intersection theory. The essential role played by
degenerate instantons is also explained. This paper is a slightly expanded
version of the author's talk at the June 1995 Trieste Conference on S-Duality
and Mirror Symmetry.
