Given a manifold M, one can study the configuration space of n points on the manifold, which is the subspace of M^n in which two points cannot be in the same position. Despite their apparent simplicity such configuration spaces are remarkably complicated; even the homology of these spaces is reasonably unknown, let alone their (rational/real) homotopy type. This classical problem in algebraic topology has much impact in more "modern" mathematics, namely in the Goodwillie-Weiss embedding calculus and in factorization homology, where one is interested not only in configuration spaces of points, but also in natural actions from the little discs operad.
In this talk, I will give an introduction to the problem of understanding configuration spaces and present a combinatorial/algebraic model of these spaces using graph complexes that, under some conditions, model the right operadic actions. We will explore some applications and see how these models allow us to answer fundamental questions about the dependence on the homotopy type of M.