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In 1983, Donaldson shocked the topology world by using instantons fromphysics to prove new theorems about four-dimensional manifolds, and hedeveloped new topological invariants. In 1988, Witten showed how theseinvariants could be obtained by correlation functions for a twisted N=2 SUSYgauge theory. In 1994, Seiberg and Witten discovered dualities for suchtheories, and in particular, developed a new way of looking at four-dimensionalmanifolds that turns out to be easier, and is conjectured to be equivalent to,Donaldson theory.  This review describes the development of this mathematical subject, and showshow the physics played a pivotal role in the current understanding of this areaof topology.Comment: 51 pages, 10 figures, 8 postscript files. Submitted to International  Journal of Modern Physics A, July 2002 Uses Latex 2e with class file  ws-ijmpa.cls (included in tar file

WHAT DO TOPOLOGISTS WANT FROM SEIBERG–WITTEN THEORY?