G protein-coupled receptor (GPCR) kinases (GRKs) and arrestins interact with agonist-bound GPCRs to promote receptor desensitization and downregulation. They also trigger signaling cascades distinct from those of heterotrimeric G proteins. Biased agonists for GPCRs that favor either heterotrimeric G protein or GRK/arrestin signaling are of profound pharmacological interest because they could usher in a new generation of drugs with greatly reduced side effects. One mechanism by which biased agonism might occur is by stabilizing receptor conformations that preferentially bind to GRKs and/or arrestins. In this review, we explore this idea by comparing structures of GPCRs bound to heterotrimeric G proteins with those of the same GPCRs in complex with arrestins and GRKs. The arrestin and GRK complexes all exhibit high conformational heterogeneity, which is likely a consequence of their unusual ability to adapt and bind to hundreds of different GPCRs. This dynamic behavior, along with the experimental tactics required to stabilize GPCR complexes for biophysical analysis, confounds these comparisons, but some possible molecular mechanisms of bias are beginning to emerge. We also examine if and how the recent structures advance our understanding of how arrestins parse the "phosphorylation barcodes" installed in the intracellular loops and tails of GPCRs by GRKs. In the future, structural analyses of arrestins in complex with intact receptors that have well-defined native phosphorylation barcodes, such as those installed by the two nonvisual subfamilies of GRKs, will be particularly illuminating.