The exchange property, introduced by Crawley and Jónsson in 1964 in the study of direct decompositions of algebraic systems and later extended to modules and rings by Warfield, plays a central role in modern decomposition theory. One of the main open problems in the area is whether the finite exchange property implies the full exchange property. This talk surveys the development of exchange theory from its module-theoretic origins to its ring-theoretic formulation via exchange rings and lifting theory.