The story of Carmen is based on the novel by Prosper Mérimée, which became famous thanks to the Bizet's opera Carmen. Interestingly, the first ballet Carmen was staged long before Bizet created his opera. Marius Petipa conceived a ballet entitled Carmen et son Toréro around the time Mérimée’s story was first published (1845). At that time, Petipa worked at the King’s Theatre in Madrid, unfortunately, there are no records left of what musical score Petipa had used.

After the appearance of the Bizet's Carmen, many ballet versions appeared: in Alhambra Theatre (London) versions by Aimé Bertrand (1897), Lucia Cormani (1903), and Augustin Berger (1912). In Chicago Guns and Castanets a version of Carmen by Metropolitan Opera ballerina Ruth Page (in 1930s).*

List of currently performed versions of the Ballet Carmen:

1949
Choreographer: Rolan Petit (France)                                                                                             
Title: Carmen
Music: Georges Bizet arranged and orchestrated by Tommy Desserre
Scenario: 1845 novel by Prosper Mérimée (France)
Acts: 3
1st Cast: Zizi Jeanmaire / Petit's wife (Carmen), Roland Petit (Don José), and Serge Perrault (Le Toréador)
Theatre: Prince's Theatre (London)

1967
Choreographer: Alberto Alonson (Cuba)                                                                                         
Title: Carmen Suite
Music: Georges Bizet arranged and orchestrated by Rodion Shchedrin
Acts: 1
1st Cast: Maya Plisetzkaya / Shchedrin's wife (Carmen), Nicolay Fadeyechev (Don José), Sergei Radchenko (Le Toréador)
Theatre/Company: Bolshoi Theatre (Moscow)

1983
Choreographer: Antonio Gades and Carlos Saura (Spain)                                                                 
Title: Carmen
Music: Antonio Gades, Solera Freire, Georges Bizet Carmen, M. Penella El gato Montesand José Ortega Heredia/Federico Garcia Lorca Verde que te quiero verde.
Scenario: inspired by the novel by Prosper Mérimée (France)
Acts: 1, duration: 80 minutes without an interval
Theatre/Company: Théâtre de Paris / Antonio Gades Company

“I did Carmen because I didn’t like the stereotyped false image of her. She’s a woman who when she loves gives herself wholly and who never forgets what class she is even in the most exalted company.” Antonio Gades

1992                                                                                                                                                                                                                                                                    Choreographer: Mats Ek (Sweden)                                                                                                
Title: Carmen
Music: Georges Bizet, Rodion Shchedrin
Costume design: Marie-Louise Ekman
Acts: 1, duration: 80 minutes without an interval                                                                                                                                                                                                             1st Cast: Ana Laguna (Carmen)
Theatre/Company: Riksteatern (Stockholm) / Cullberg ballet

2007
Choreographer: Roland Petit (France)                                                                                            
Title: Carmen. Solo
Music: Georges Bizet
Acts: 1
1st Cast: Nikolai Tsiskaridze (for the project "Kings of the dance")

2014
Choreographer: Dada Masilo (South Africa)                                                                                    
Title: Carmen
Music: Georges Bizet, Rodion Chtchedrin, Arvo Pärt
Costume Design: Dada Masilo et Suzette Le Sueur
Acts: 4
Cast & Details: Piece for 14 dancers (including Dada Masilo), run time 1h10 approx Theatre/ Company: Maison de la Danse, Biennale de la danse / The Dance Factory (Direction by Suzette Le Sueur)

"Carmen is more in the hips — rough, sexual, sensual. It’s not only about telling the story but about finding the movement vocabulary for it. You have to find the emotion in the body, physically. I want to bring realism into the production. There are quite a lot of fight scenes in Carmen and I didn’t want to choreograph them. I want things to be real and a little scary for the viewer. We’re also being more vocal with Carmen now because it is a dramatic story. I have found with dance there’s a point at which you can’t move anymore, and you have to vocalise an emotion. When we did this in Europe, they were amazed. But for me it’s about working with, and taking from, the edge and energy of Johannesburg, the edge of my roots and where I come from.” Dada Masilo

2015
Choreographer: Carlos Acosta (Cuba)                                                                                           
Title: Carmen
Music: Georges Bizet, orchestration by Martin Yates
Designer: Tim Hatley (Central Saint Martins graduate)
Acts: 1, approx.60 min
1st Cast: Marianela Nuñez (Carmen), Carlos Acosta (Don José), Federico Bonelli (Le Toréador)
Theatre/Company: Royal Opera House

Sources:

(1) website en scenes le spectacle vivant en video 

(2) article on wikipedia

(3) website of the Antonio Gades Foundation 

(4) website Dance from Spain created by FECED

(5) website of the 16th biennale de la danse de lyon 

(6) Mail & Guardian Online 

(7) blog a dance history

(8) article Royal Ballet mixed bill review – Carlos Acosta throws too much at Carmen from The Guardian, by Judith Mackrell, 2015

(9) Royal Opera House web-site

(10) *The ballet blog