  We explore the possibility of generalizing the locally localized gravity
model in five space-time dimensions to arbitrary higher dimensions. In a
space-time with negative cosmological constant, there are essentially two kinds
of higher-dimensional cousins which not only take an analytic form but also are
free from the naked curvature singularity in a whole bulk space-time. One
cousin is a trivial extension of five-dimensional model, while the other one is
in essence in higher dimensions. One interesting observation is that in the
latter model, only anti-de Sitter ($AdS_p$) brane is physically meaningful
whereas de Sitter ($dS_p$) and Minkowski ($M_p$) branes are dismissed.
Moreover, for $AdS_p$ brane in the latter model, we study the property of
localization of various bulk fields on a single brane. In particular, it is
shown that the presence of the brane cosmological constant enables bulk gauge
field and massless fermions to confine to the brane only by a gravitational
interaction. We find a novel relation between mass of brane gauge field and the
brane cosmological constant.
