The set of all rearrangement invariant function spaces on [0,1] having the p-Banach–Saks property has a unique maximal element for all p∈(1,2]. For p=2 this is L 2, for p∈(1,2) this is L p,∞ 0. We compute the Banach–Saks index for the families of Lorentz spaces L p,q, 1<p<∞ , 1⩽ q⩽∞, and Lorentz–Zygmund spaces L( p, α), 1⩽p<∞, α∈ R , extending the classical results of Banach–Saks and Kadec–Pelczynski for L p -spaces. Our results show that the set of rearrangement invariant spaces with Banach–Saks index p∈(1,2] is not stable with respect to the real and complex interpoltaion methods. To cite this article: E.M. Semenov, F.A. Sukochev, C. R. Acad. Sci. Paris, Ser. I 337 (2003).