1953 in reference to experimental theater productions in New York City, from off (prep.) + Broadway. Even more experimental off-off-Broadway is attested by 1958.
Entries linking to off-broadway
off prep., adv.
by c. 1200 as an emphatic form of Old English of (see of), employed in the adverbial use of that word. The prepositional meaning "away from" and the adjectival sense of "farther" were not firmly fixed in this variant until 17c., but once they were they left the original of with the transferred and weakened senses of the word. Meaning "not working" is from 1861.
Off the cuff "extemporaneously, without preparation" (1938) is from the notion of speaking from notes written in haste on one's shirt cuffs. In reference to clothing, off the rack (adj.) "not tailored, not made to individual requirements, ready-made" is by 1963, on the notion of buying it from the rack of a clothing store; off the record "not to be publicly disclosed" is from 1933; off the wall "crazy" is 1968, probably from the notion of a lunatic "bouncing off the walls" or else in reference to carom shots in squash, handball, etc.
Broadway
common street name, c. 1300 as "a wide road or street," from broad (adj.) + way (n.); the allusive use for "New York theater district" is by 1881.
Obie
one of the annual awards given to off-Broadway theater, 1967, from O.B. as the abbreviation of Off-Broadway.