But now the Sharks and the Jets - the show's rival New York gangs - are finally getting some new moves.
Manchester's Royal Exchange theatre is to stage what's thought to be the first full UK version with new choreography.
And another new production, also with new choreography, is going to Broadway.
That version, which opens in 2020, will be the first major New York revival to deviate significantly from Robbins' moves.
It will be directed by Ivo Van Hove and choreographed by Anne Teresa De Keersmaeker. Meanwhile, Steven Spielberg is also directing a new film version.
Music Theatre International, which controls the rights for the show, said the Manchester production, which will open in April 2019, is believed to be the first to do so in the UK.
West Side Story, which transfers the story of Romeo and Juliet to the Big Apple, first opened on Broadway in 1957, and was turned into a landmark film four years later, winning 10 Oscars.
'Daunting' Robbins' dance moves have been seen as sacrosanct since then.
The new choreography for the Manchester show will be created by Aletta Collins, a former associate artist at the Royal Opera House.
She said it was "daunting" to be reimagining the show's moves, but it was "an honour to have that opportunity".
Part of the reason the show is being reimagined for Manchester is that the Royal Exchange is a theatre in the round.
"I think I'm very lucky that I'm not coming to the piece in a formal proscenium arch space [a traditional theatre layout], as the original production was conceived," Collins said.
"Actually, having this challenge of the round will immediately make us look at different ways of creating this world.
"So actually it will be fresh and new and all those exciting things we want a new production to be, but in dialogue with the space in the round.
"One physically can't reproduce that original iconic choreography that I've grown up with. So it's certainly a challenge, and I'm very happy to embrace it."