Tout commence au début des années 1970 ; à Londres, la comédie musicale Jesus Christ Superstar de Andrew Lloyd Webber et Tim Rice connaît un grand succès. Un soir, un des figurants est renvoyé: il s'agit de Richard O'Brien. Ayant une femme et un enfant, et disposant de pas mal de temps libre, il décide d'écrire sa propre comédie musicale. Il se base sur ses goûts et ses souvenirs d'adolescents, et décide d'écrire une riposte aux grands spectacles qui se jouaient alors sur la scène londonienne en prenant comme thèmes majeurs de cet « anti-spectacle » le sexe, les films de série B et le rock and roll.
Le jour de la première, le 19 juin 1973, il pleut et seules quelques dizaines de spectateurs sont là pour voir Richard O'Brien et Tim Curry, qui interprètent respectivement les rôles de Riff Raff et de Frank-n-Furter. L'année suivante, le spectacle est le "Meilleur Spectacle de l'année 1974".
Malgré un relatif échec à Broadway, la pièce se jouera sans interruption à Londres jusqu'en 1980. Elle est toujours jouée aujourd'hui à travers le monde, de deux façons différentes: soit intégralement interprétée par des acteurs sur scène, soit avec projection du film agrémenté d'acteurs apparaissant devant l'écran pour rejouer certaines scènes cultes ou interagir avec les images. Cette dernière forme de prestation impliquant le public a contribué à faire du Rocky Horror Show un spectacle culte.
The Usherette in a derelict cinema introduces tonight's film in a song ("Science Fiction/Double Feature"), with masked Phantoms providing the backing vocals.
After attending the wedding of Ralph Hapshatt and Betty Munroe, Brad Majors confesses his love to Janet Weiss ("Dammit Janet") and the two become engaged. The Narrator appears and explains that Brad and Janet are leaving Denton to visit Dr. Everett Scott, their former science tutor, while driving into a rainstorm. During the trip, their car has a flat tire and they are forced to walk through the rain to seek a telephone in an old castle ("Over at the Frankenstein Place").
The Narrator explains that Brad and Janet are feeling "apprehensive and uneasy," but must accept any help that they are offered. As Brad and Janet arrive, Riff Raff, the hunchbacked handyman, greets them, and his sister Magenta, the maid, appears. Riff Raff, Magenta and Columbia (a groupie) speak briefly of an unlucky delivery boy named Eddie before performing the show's signature dance number ("Time Warp"*). Brad and Janet try to leave at this point, but are stopped when Dr. Frank N. Furter, a pansexual, cross-dressing mad scientist, arrives. He introduces himself as "a sweet transvestite from Transsexual, Transylvania" and invites Brad and Janet up to his laboratory ("Sweet Transvestite"). As he goes up, Brad and Janet are stripped to their underwear to dry off.
Brad and Janet enter the laboratory, where Frank N. Furter gives them laboratory coats to wear. Frank announces that he has discovered the secret to life itself. He unveils his creation, a blonde, well-built man named Rocky Horror, who is brought to life. As his bandages are removed, Rocky worries about his predicament ("The Sword of Damocles"). Frank admires Rocky's physique by singing a tribute to muscle builders ("Charles Atlas Song"/"I Can Make You a Man"**). A Coca-Cola freezer in the laboratory opens to reveal Frank and Columbia's former lover, Eddie, a biker covered in surgical scars. Eddie — now rendered a zombie after having part of his brain removed by Frank — contemplates his new existence and recalls his former life ("Hot Patootie – Bless My Soul"). Frank panics, forces Eddie back into the freezer and hacks him to death. Frank tells Rocky — the recipient of the other half of Eddie's brain — that he prefers him over Eddie ("Charles Atlas Song (Reprise)"/"I Can Make You a Man (Reprise)"**). Brad and Janet are then ushered to separate bedrooms for the night.
The Narrator foreshadows that Brad and Janet may be quite unsafe. Janet enjoys Brad's advances in her darkened bedroom before realizing that it is in fact Frank in disguise. He convinces Janet that pleasure is no crime, and after she asks him to promise not to tell Brad, they resume their lovemaking. The scene changes to Brad's darkened bedroom, where Brad makes love to Janet before discovering that, once again, it is Frank in disguise. Frank promises not to tell Janet, but as they resume, Riff Raff interrupts on the television monitor with the message that Rocky has escaped. Janet searches for Brad in the laboratory and discovers Rocky hiding there. Checking the television monitor, Janet sees Brad in bed with Frank and seduces Rocky ("Touch-a, Touch-a, Touch-a, Touch Me"). While searching the television monitor for Rocky, the rest of the group discovers that Janet has slept with him and Brad becomes hurt and angry ("Once in a While"). Riff Raff then notifies Frank that there is another visitor entering the castle: Doctor Everett Scott, the paraplegic science tutor whom Brad and Janet intended to visit.
Doctor Scott is pushed into the laboratory by Columbia where Frank accuses him and Brad of trying to investigate his castle, knowing that Doctor Scott has connections with the FBI. Doctor Scott assures him that he has come in search of Eddie, who is revealed to be his nephew ("Eddie's Teddy"). Frank displays Eddie's corpse to the group and then uses a device to electronically restrain the three visitors and a rebellious Rocky to the floor ("Planet Schmanet Janet"); the inhabitants of the castle are revealed to be space aliens led by Frank, who abandoned their original mission in order to engage in kinky sex with earthlings and work on Rocky. Magenta insists that they return to their home planet now that they have been found out; Frank refuses and, instead, declares his intentions to put on a "floor show."
Under Frank's influence, Columbia, Rocky, Brad, and Janet perform song and dance routines while clad in lingerie ("Rose Tint My World"). After, Frank entices them to lose all inhibition and give in to their basest carnal instincts, resulting in everyone's beginning to engage in orgiastic sex ("Don't Dream It – Be It") before they are interrupted by Frank, who leads them into the concluding number of the floor show ("Wild and Untamed Thing"). The show comes to an abrupt end when Riff Raff and Magenta enter, wearing spacesuits and carrying ray guns. Riff Raff declares that he is usurping Frank's authority and taking them all back to their home planet. Frank makes a final plea for sympathy from Riff Raff, trying to make him understand his desire to spend the rest of his life having sex with earthlings ("I'm Going Home"). Riff Raff is unmoved and guns down Columbia, Frank and Rocky before ordering Brad, Janet and Doctor Scott to leave.
As the trio evacuates the castle, Riff Raff and Magenta express their excitement to return to their world and do the "Time Warp" again with their fellow Transylvanians ("Spaceship"). Brad and Janet watch as the castle blasts off into outer space, confused about the implications of their sexual escapades ("Super Heroes"). To conclude his tale, the Narrator says "and crawling on the planet's face, insects called the human race, lost in time, and lost in space – and meaning." As the show ends, The Usherette returns to recount the night's events ("Science Fiction/Double Feature (Reprise)").
* In the original London and Los Angeles productions, "Sweet Transvestite" came before "Time Warp". This was changed for the film version and was subsequently updated for the stage version when Richard O'Brien revised the script for the 1990 West End revival.
* "Charles Atlas Song" was replaced by a reworked version of the song, "I Can Make You a Man", for the film version. Richard O'Brien's revision of the script in 1990 featured a hybrid of the two songs under the title "I Can Make You a Man", in the 1999 revised script this song was replaced by the film version which continues to be used in all major productions. The reprise remains unchanged except for the title.