We solve the Landau problem for charged particles on odd dimensional spheres S2k-1 in the background of constant SO(2k - 1) gauge fields carrying the irreducible representation (I/2,I/2, . . . , I/2). We determine the spectrum of the Hamiltonian, the degeneracy of the Landau levels and give the eigenstates in terms of the Wigner D-functions, and for odd values of I, the explicit local form of the wave functions in the lowest Landau level (LLL). The spectrum of the Dirac operator on S2k-1 in the same gauge field background together with its degeneracies is also determined, and in particular, its number of zero modes is found. We show how the essential differential geometric structure of the Landau problem on the equatorial S2k-2 is captured by constructing the relevant projective modules. For the Landau problem on S-5, we demonstrate an exact correspondence between the union of Hilbert spaces of LLLs, with I ranging from 0 to I-max = 2K or I-max = 2K or I-max = 2K + 1 to the Hilbert spaces of the fuzzy CP3 or that of winding number +/- 1 line bundles over CP3 at level K, respectively.