For many known families of algebro-geometric objects indexed by natural numbers (symmetric groups, braid groups, ...), a phenomenon known as homological stability happens: their homology H_d(X_n) becomes stationary when n goes to infinity. Sometimes, these objects are equipped with actions of the symmetric group \Sigma_n on X_n: in this context, the right notion of stability is that of representation stability introduced by Church and Farb in 2010. One main example is the family of ordered configuration spaces of manifolds, whose cohomology is known to be representation stable with explicit ranges. I will review in this talk the main ideas of representation stability and explain how we can take a homotopy-theoretic point of view on representation stability to generalize such stability theorems. (This talk is part of the activity of the MIUR Excellence Department Projects MatMod@TOV.)