  The study of the Banach-Saks property in Banach spaces has a long and
illustrious history. Of late, motivated by applications in financial
mathematics, interest has arisen in the Banach-Saks type properties with
respect to order convergence. This paper presents a study of order Banach-Saks
properties in Banach function spaces, and in particular in rearrangement
invariant spaces. Among the results obtained, we provide some sufficient
conditions for the (weak) order Banach-Saks property. We also characterize the
(weak) order Banach-Saks property in Orlicz spaces. It is also shown that the
(weak) order Banach-Saks property is equivalent to its hereditary version.
