Artin groups are a generalization of braid groups, and arise as the fundamental groups of configuration spaces associated with Coxeter groups. A long-standing open problem, called the $K(\pi,1)$ conjecture, states that the higher homotopy groups of these configuration spaces are trivial. For finite Coxeter groups, this was proved by Deligne in 1972. In the first part of this talk, I will introduce Coxeter groups, Artin groups, and the $K(\pi,1)$ conjecture (so that only a few topological and combinatorial prerequisites are needed). Then I will outline a recent proof of the $K(\pi,1)$ conjecture in the affine case, which is joint work with Mario Salvetti.