The addition of a tail of polyadenylic acid (POLY A) to the 3' end of mRNA (RNA, MESSENGER). Polyadenylation involves recognizing the processing site signal, (AAUAAA), and cleaving of the mRNA to create a 3' OH terminal end to which poly A polymerase (POLYNUCLEOTIDE ADENYLYLTRANSFERASE) adds 60-200 adenylate residues. The 3' end processing of some messenger RNAs, such as histone mRNA, is carried out by a different process that does not include the addition of poly A as described here.