Configuration spaces consist of ordered collected ofpairwise distinct points in a given manifold. In this talk, I will present several algebraic models for the real/rational homotopy types of (possibly framed) configuration spaces. These models canbe used to establish real/rational homotopy invariance of configuration spaces under dimensionality and connectivity assumptions. Moreover, the collection of all configuration spacesof a given manifold has the structure of a right module over some version of the little disks operad, and the algebraic models are compatible with this extra structure. The proofs all use ideas from the theory of operads, namely Kontsevich's proof of the formality of the little disks operad and – for oriented surfaces – Tamarkin's proof of the formality of the little 2-disks operad.(Based on joint works with Campos, Ducoulombier, Lambrechts, and Willwacher.)