This article describes the use of symplectic cut-and-paste methods to compute Gromov-Witten invariants. Our focus is on recent advances extending these methods to Kähler surfaces with geometric genus pg > 0, for which the usual GW invariants vanish for most homology classes. This involves extending the Splitting Formula and the Symplectic Sum Formula to the family GW invariants introduced by the first author. We present applications the invariants of elliptic surfaces and to the Yau-Zaslow

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... cture. In both cases the results agree with the conjectures of algebraic geometers and yield a proof (to appear [LL]) of previously unproved cases of the Yau-Zaslow Conjecture.