Let　H　be　a　k-partite　k-graph　with　n　vertices　in　each　partition　class,　and　let　delta(k-1)　(H)　denote　the　minimum　codegree　of　H.　We　characterize　those　H　with　delta(k-1)　(H)　＞=　n/2　and　with　no　perfect　matching.　As　a　consequence,　we　give　an　affirmative　answer　to　the　following　question　of　Rodl　and　Rucinski:　if　k　is　even　or　n　not　equivalent　to　2　(mod　4),　does　delta(k-1)　(H)　＞=　n/2　imply　that　H　has　a　perfect　matching?　We　also　give　an　example　indicating　that　it　is　not　sufficient　to　impose　this　degree　bound　on　only　two　types　of　(k　-　1)-sets.