CRISPR-associated Rossmann Fold (CARF) and SMODS-associated and fused to various effector domains (SAVED) are key components of cyclic oligonucleotide-based antiphage signaling systems (CBASS) that sense cyclic oligonucleotides and transmit the signal to an effector inducing cell dormancy or death. Most of the CARFs are components of a CBASS built into type III CRISPR-Cas systems, where the CARF domain binds cyclic oligoA (cOA) synthesized by Cas10 polymerase-cyclase and allosterically activates the effector, typically a promiscuous ribonuclease. Additionally, this signaling pathway includes a ring nuclease, often also a CARF domain (either the sensor itself or a specialized enzyme) that cleaves cOA and mitigates dormancy or death induction. We present a comprehensive census of CARF and SAVED domains in bacteria and archaea, and their sequence- and structure-based classification. There are 10 major families of CARF domains and multiple smaller groups that differ in structural features, association with distinct effectors, and presence or absence of the ring nuclease activity. By comparative genome analysis, we predict specific functions of CARF and SAVED domains and partition the CARF domains into those with both sensor and ring nuclease functions, and sensor-only ones. Several families of ring nucleases functionally associated with sensor-only CARF domains are also predicted.