abstract: Residual torsion-free nilpotence is the strongest of the usual residual properties which can be expected to hold for the fundamental group of the complement of an arbitrary hyperplane arrangement. Is is known to hold for supersolvable or ber-type arrangements thanks to the work of Falk and Randell. This case includes the most standard reflection arrangements, but leaves open the case of many others. I will present advances and evidences toward a positive answer to the general problem, in the case of reflection arrangements.