Roughly speaking, the Gromov-Witten invariants of a
symplectic manifold count how many holomorphic curves it has. In this
talk we describe a gluing formula for the Gromov-Witten invariants of a
symplectic sum of X and Y along V in terms of the relative
Gromov-Witten invariants of the pairs (X,V) and (Y,V). Our formula
(joint work with Tom Parker) describes what happens to holomorphic
curves as one degenerates the syplectic sum into the union of X and Y along
V and explains how to compute the GW invariant from the limiting curves.
 
Applications of this formula range from constructions of exotic symplectic
manifolds, to recursive formulas for the  GW invariants, including new
relations in the cohomology ring of the moduli space of complex structures
on a genus g Riemann surface with n marked points.