Acts as the catalytic component of the CCR4-NOT core complex, which in the nucleus seems to be a general transcription factor, and in the cytoplasm the major mRNA deadenylase involved in mRNA turnover. In vivo and in vitro, caf1 has 3'-exoribonuclease activity with a preference for poly(A) RNAs. Exonucleolytic cleavage of poly(A) to 5'-AMP. Binds 1 Mn(2+) ion per subunit. Binds 1 Zn(2+) ion per subunit. At lower physiological Zn(2+) concentrations, Mg(2+) replaces Zn(2+). The nature of the bound ion affects the speed of the RNA degradation reaction and, to a limited extent, base selectivity. Belongs to the CAF1 family.