The process by which the 3' ends of most eukaryotic mrnas are formed; the covalent modification of a macromolecule (e.g., mrna) by the formation of a polyadenyl moiety covalent linked to the macromolecule; post-transcriptional addition of a polyadenylic acid tail to the 3' end of eukaryotic mrnas; also called 'poly-(A) tailing'.