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Consider an affine Coxeter group $W$ acting by isometries on the Euclideanspace $\mathbb{R}^n$, and the arrangement of its reflection hyperplanes. Thefundamental group of the complement $Y_W$ of the complexification of thisarrangement in $\mathbb{C}^n$ mod out by $W$ is the affine Artin group $G_W$associated with $W$. The $K(\pi,1)$ conjecture states that $Y_W$ is aclassifying space for $G_W$. It has been recently proved by Paolini andSalvetti building on the works of McCammond and Sulway. We will present someingredients of the proof that rests on the study of dual Garside structures foraffine Artin groups, the factorisations of Euclidean isometries, and theshellability of noncrossing partitions. One consequence is that affine Artingroups, as well as braided crystallographic groups, have a finite classifyingspace.

La conjecture du $K(\pi,1)$ pour les groupes d'Artin affines (d'apr\`es Paolini et Salvetti)