Linkage-specific polyubiquitin recognition is thought to make possible the diverse set of functional outcomes associated with ubiquitination. Thus far, mechanistic insight into this selectivity has been largely limited to single domains that preferentially bind to lysine 48-linked polyubiquitin (K48-polyUb) in isolation. Here, we propose a mechanism, linkage-specific avidity, in which multiple ubiquitin-binding domains are arranged in space so that simultaneous, high-affinity interactions are optimum with one polyUb linkage but unfavorable or impossible with other polyUb topologies and monoUb. Our model is human Rap80, which contains tandem ubiquitin interacting motifs (UIMs) that bind to K63-polyUb at DNA double-strand breaks. We show how the sequence between the Rap80 UIMs positions the domains for efficient avid binding across a single K63 linkage, thus defining selectivity. We also demonstrate K48-specific avidity in a different protein, ataxin-3. Using tandem UIMs, we establish the general principles governing polyUb linkage selectivity and affinity in multivalent ubiquitin receptors.