Quantum Hall effects on the complex Grassmann manifolds Gr(2)(C-N) are formulated. We set up the Landau problem in Gr(2()C(N)) and solve it using group theoretical techniques and provide the energy spectrum and the eigenstates in terms of the SU(N) Wigner D functions for charged particles on Gr(2()C(N)) under the influence of Abelian and non-Abelian background magnetic monopoles or a combination of these. In particular, for the simplest case of Gr(2()C(4)), we explicitly write down the U(1) background gauge field as well as the single- and many-particle eigenstates by introducing the Plucker coordinates and show by calculating the two-point correlation function that the lowest Landau level at filling factor v=1 forms an incompressible fluid. Our results are in agreement with the previous results in the literature for the quantum Hall effect on CPN and generalize them to all Gr(2()C(N)) in a suitable manner. Finally, we heuristically identify a relation between the U(1) Hall effect on Gr(2()C(4)) and the Hall effect on the odd sphere S-5, which is yet to be investigated in detail, by appealing to the already-known analogous relations between the Hall effects on CP3 and CP7 and those on the spheres S-4 and S-8, respectively.