The Bernstein-Sato polynomial is a subtle invariant of a hypersurface singularity that is closely related to the log-canonical threshold. The topological zeta function is another invariant, and the strong topological monodromy conjecture of Denef-Loeser relates these two. In this talk I will present a proof of this conjecture for the case of Weyl hyperplane arrangements, namely those arising from root systems of simple Lie algebras. I will also present an upper bound and conjectured formula for the arrangements of type A_n. The talk is based on joint work with Robin Walters.