The associahedra, introduced by Stasheff in his study of loop spaces, are now a classical sequence of polytopes, related to homotopy associativity and moduli spaces. Another sequence of related polytopes, called cyclohedra, have been introduced later in relation with knots.

Recently, S. Fomin and A. Zelevinsky have associated to any finite root system a complete simplicial fan, which has for type A the combinatorics of associahedra and for type B the combinatorics of cyclohedra. They conjectured that these fans are the normal fans of simple convex polytopes.

This conjecture has now been proved in a joint work with S. Fomin and A. Zelevinsky. We explain the initial motivation, detail the combinatorics of the fan and give a sketch of the proof of the polytopality conjecture.