  We extend the recent D=5 results of Dias, Horowitz and Santos by finding
asymptotically AdS rotating black hole and boson star solutions with scalar
hair in arbitrary odd spacetime dimension. Both the black holes and the boson
stars are invariant under a single Killing vector field which co-rotates with
the scalar field and, in the black hole case, is tangent to the generator of
the horizon. Furthermore, we explicitly construct boson star and small black
hole ($r_+ \ll \ell$) solutions perturbatively assuming a small amplitude for
the scalar field, resulting in solutions valid for low energies and angular
momenta. We find that just as in D=5, the angular momentum is primarily carried
by the scalar field in $D>5$, whereas unlike D=5 the energy is also primarily
carried by the scalar field in $D>5$; the thermodynamics in D=5 are governed by
both the black hole and scalar field whereas in $D>5$ they are governed
primarily by the scalar field alone. We focus on cataloguing these solutions
for the spacetime dimensions of interest in string theory, namely $D=5,7,9,11$.
