The CUL4A-RBX1-DDB1-DDB2 ligase (CRL4ADDB2) is the primary DNA damage-sensing complex in the repair of UV light induced pyrimidine dimers1. The class of cullin-RING ubiquitin ligases (CRLs) comprises six canonical cullin families (Cul1, Cul2, Cul3, Cul4A, Cul4B, and Cul5), which together with the RING-domain proteins Rbx1 or Rbx2 mediate ubiqutination in ~20% of the proteins degraded by the proteasome2. CRLs assembly is modular, with sets of receptors and adaptors producing hundreds of distinct cullin-RING E3 ubiquitin ligase complexes3. Regulation of this structurally diverse family of ligases depends on the COP9 signalosome (CSN), an 8-subunit isopeptidase that removes the covalently conjugated Nedd8 activator from the cullin4-6. In addition to its catalytic function, CSN form tight complexes with the unmodified CRL, a process implicated in preventing the CRL substrate adaptor from undergoing cycles of futile auto-ubiquitination and degradation in vivo7-9. CSN inhibition is overcome once the CRL binds a substrate. Here we present the structure of CRL4 and a dimeric CRL3 in complex with CSN using single-particle electron microscopy (EM). We find that the conserved cullin C-terminus together with Rbx1 is held by CSN2 and CSN4 subunits. In the CRL1, CRL3 and CRL4 families, the divergent receptors and adaptors form a variety of contacts that are unexpectedly plastic in nature involving CSN1, CSN3 and CSN helical bundle. Altogether our findings imply a model where steric repulsion by the cognate substrate allow CSN-mediated regulation of ~300 different CRL enzymes in response to different cues, without the need for dedicated interactions or common motifs.