A classical result usually attributed to Cayley and Bacharach asserts that if two plane curves of degrees c and d meet in cd points, then any curve of degree (c + d - 3) passing through all but one of these points must also pass through the remaining one. In the late 1970s, Griffiths and Harris showed that this is a special case of a general result about zero-loci of sections of a vector bundle. Inspired by a recent paper of Mu-Lin Li, I will describe a generalization allowing for excess vanishing. Multiplier ideals enter the picture in a natural way. Time permitting, I will also explain how a result due to Tan and Viehweg leads to statements of Cayley-Bacharach type for determinantal loci. This is joint work with Lawrence Ein.[-]