Acts as 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 (By similarity). Ccr4 has 3'-5' RNase activity with a strong preference for polyadenylated substrates and also low exonuclease activity towards single-stranded DNA (By similarity). Exonucleolytic cleavage of poly(A) to 5'-AMP. Belongs to the CCR4/nocturin family.