Linkage, or liaison, is a tool for classifying and studying varieties and ideals that has its origins in 19th century algebraic geometry. Its generalization, residual intersection, has broad applications in enumerative geometry, intersection theory, the study of Rees rings, and multiplicity theory. After surveying basic properties of linkage, we will focus on the computation of Picard groups and divisor class groups and on the structure of rigid algebras in the linkage class of a complete intersection. We will describe applications of residual intersections and explain the techniques used to determine their Cohen-Macaulayness, canonical modules, duality properties, and defining equations. An emphasis will be on weakening the hypotheses classically required in this subject.