  The noncommutative harmonic oscillator in arbitrary dimension is examined. It
is shown that the $\star$-genvalue problem can be decomposed into separate
harmonic oscillator equations for each dimension. The noncommutative plane is
investigated in greater detail. The constraints for rotationally symmetric
solutions and the corresponding two-dimensional harmonic oscillator are solved.
The angular momentum operator is derived and its $\star$-genvalue problem is
shown to be equivalent to the usual eigenvalue problem. The $\star$-genvalues
for the angular momentum are found to depend on the energy difference of the
oscillations in each dimension. Furthermore two examples of assymetric
noncommutative harmonic oscillator are analysed. The first is the
noncommutative two-dimensional Landau problem and the second is the
three-dimensional harmonic oscillator with symmetrically noncommuting
coordinates and momenta.
