In this talk, we give the formulation of Quantum Hall Effects (QHEs) on the complex Grassmann manifolds Gr(2)(C-N). We set up the Landau problem in Gr(2)(C-N), 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 thereof. For the simplest case of Gr(2)(C-4) we provide explicit constructions of the single and many-particle wavefunctions by introducing the Plucker coordinates and show by calculating the two-point correlation function that the lowest Landau level (LLL) at filling factor nu = 1 forms an incompressible fluid. 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. The talk is given by S. Kurkcuoglu at the Group 30 meeting at Ghent University, Ghent, Belgium in July 2014 and based on the article by F.Balh, A.Behtash, S.Kurkcuoglu, G.Unal .