The posttranslational conversion of C-terminal glycine-extended peptides to C-terminal alpha-amidated peptides. Occurs to over half of all peptide hormones to give bioactive peptides. This is a two step process catalyzed by a peptidyl-glycine alpha-hydroxylating monooxygenase and a peptidyl-alpha-hydroxyglycine alpha-amidating lyase. In some organisms, this process is catalyzed by two separate enzymes, whereas in higher organisms, one polypeptide catalyzes both reactions.