This paper considers invariant almost Hermitian structures on a flag manifold G/P = U/K where G is a complex semi-simple Lie group, P is a parabolic subgroup of G, U is a compact real form of G and K = U boolean AND P is the centralizer of a torus. The main result shows that there are nearly-Kahler structures in G/P which are not Kahler if and only if G/P has height two. This proves for the flag manifolds a conjecture by Wolf and Gray.1211143154