early 14c., wrappen, "to wind (something around something else), cover (something), conceal; bind up, swaddle; fold (something) up or back on itself," of uncertain origin, perhaps via Scandinavian (compare Danish dialectal vravle "to wind"), from PIE *werp- "to turn, wind," from root *wer- (2) "to turn, bend." Or perhaps a variant of lap (v.2). To wrap up "put an end to" is from 1926. Related: Wrapped; wrapping. Wrapping paper is from 1715.
late 15c., "fine cloth used as a cover or wrapping for bread," from wrap (v.). As a type of women's garment, recorded from 1827. Meaning "plastic film or cellophane used as a wrap" is from 1930. Meaning "end of a filming session" is attested from 1970. Meaning "sandwich material folded up in flour tortilla" is by 1998. Figurative phrase under wraps "in concealment" is recorded from 1939.