Abstract.In this paper we prove that, for 3 < k < n -3 , none of the oriented Grassmann manifolds, Gn k-except for G6 3 , and a few as yet undecided cases-admits a weakly almost complex structure. The result for k = 1,2, n -1, «-2 are well known and classical. The proofs make use of basic concepts in iT-theory, the property that Gn k is (n -fc)-universal, known facts about K(RP ), and characteristic classes.