  We consider spherically symmetric higher-dimensional solutions of Einstein's
equations with a bulk cosmological constant and n transverse dimensions. In
contrast to the case of one or two extra dimensions we find no solutions that
localize gravity when $n\geq 3$, for strictly local topological defects. We
discuss global topological defects that lead to the localization of gravity and
estimate the corrections to Newton's law. We show that the introduction of a
bulk ``hedgehog'' magnetic field leads to a regular geometry and localizes
gravity on the 3-brane with either a positive, zero or negative bulk
cosmological constant. The corrections to Newton's law on the 3-brane are
parametrically the same as for the case of one transverse dimension.
