Let \(A\) be either a nonsingular affine variety or a projective space \(\mathbb P^n\) over the field of complex numbers. Two closed subvarieties of \(A\) are said to be geometrically linked if they have no common component and their union is a complete intersection in \(A\). If one of the varieties, say \(X\), is fixed and if the complete intersection is chosen in as general a way as possible containing \(X\), then the complement \(Y\) is called a generic link of \(X\). The object of this paper is to study how singularities behave under generic linkage. The author gives a description of the Grauert-Riemenschneider canonical sheaf of \(Y\) in terms of the multiplier ideal sheaves associated to \(X\), and uses it to study the singularities of \(Y\). His work generalizes previous results of Chardin and Ulrich, among others. He gives several applications of his main theorem, for instance to the study of rational singularities and to the study of long canonical threshold under generic linkage. Finally, he applies it to generalize known results by T. De Fernex and L. Ein [Am. J. Math. 132, No. 5, 1205–1221 (2010; Zbl 1205.14020)], and by M. Chardin and B. Ulrich [Am. J. Math. 124, No. 6, 1103–1124 (2002; Zbl 1029.14016)], on the Castelnuovo-Mumford regularity bound for a projective variety.