The classical theorem of Dunford and Pettis identifies the bounded, uniformly integrable subsets of L1(μ) with the relatively weakly compact sets. Another characterization of uniform integrability is given in a theorem of De La Vallée Poussin which states that asubset K of L1(μ) is bounded and uniformly integrable if and only if there is an N-function F so thatsup{f F(f)dμ: f ε K} < ∞. De La Vallée Poussin's theorem is the focal point of the fmt part of this paper as well as the driving force for the results in the second part. We refine and improve this theorem in several directions. The theorem of De La Vallée Poussin does not, for instance, specify just how well the functionFcan be chosen. It gives little additional information in case the set in question is relatively norm compact inL1(μ).Finally it gives no information on the structure of the set in the corresponding Band space of F-integrable functions. More specifically we establish the fact that asubset K of L1is relatively compact if and only if there is an N-function Fε δ'so that K is relatively compact in L*F.Furthermore we prove thata subset K of L1is relatively weakly compact if and only if there is an N-function Fε δ'so that K is relatively weakly compact in L*F.We then go on to show that a large class of non-reflexive Orlicz spaces has the weak Band-Saks property, by establishing a result for these spaces, very similar to the Dunford-Pettis theorem forL1.