  We solve the Landau problem for charged particles on odd-dimensional spheres
$S^{2k-1}$ in the background of constant SO(2k-1) gauge fields carrying the
irreducible representation $\left ( \frac{I}{2}, \frac{I}{2}, \cdots,
\frac{I}{2} \right)$. We determine the spectrum of the Hamiltonian, the
degeneracy of the Landau levels and give the eigenstates in terms of the Wigner
${\cal D}$-functions, and for odd values of $I$ the explicit local form of the
wave functions in the lowest Landau level (LLL). Spectrum of the Dirac operator
on $S^{2k-1}$ in the same gauge field background together with its degeneracies
is also determined and in particular the number of zero modes is found. We show
how the essential differential geometric structure of the Landau problem on the
equatorial $S^{2k-2}$ 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 LLL's with $I$ ranging
from $0$ to $I_{max} = 2K$ or $I_{max} = 2K+1$ to the Hilbert spaces of the
fuzzy ${\mathbb C}P^3$ or that of winding number $\pm1$ line bundles over
${\mathbb C}P^3$ at level $K$, respectively.
