  We study the phenomenology resulting from backgrounds of the form $AdS_5
\times {\cal M}^\delta$, where ${\cal M}^\delta$ denotes a generic manifold of
dimension $\delta \geq 1$, and $AdS_5$ is the slice of 5-dimensional anti-de
Sitter space which generates the hierarchy in the Randall-Sundrum (RS) model.
The $\delta$ additional dimensions may be required when the RS model is
embedded into a more fundamental theory. We analyze two classes of
$\delta-$dimensional manifolds: flat and curved geometries. In the first case,
the additional flat dimensions may accommodate localized fermions which in turn
could resolve issues, such as proton decay and flavor, that were not addressed
in the original RS proposal. In the latter case, the positive curvature of an
$S^\delta$ manifold with $\delta> 1$ can geometrically provide the
5-dimensional warping of the RS model. We demonstrate the key features of these
two classes of models by presenting the background solutions, the spectra of
the Kaluza-Klein (KK) gravitons, and their 4-dimensional couplings, for the
sample manifolds $S^1/Z_2$, $S^1$, and $S^2$. The resulting phenomenology is
distinct from that of the original RS scenario due to the appearance of a
multitude of new KK graviton states at the weak scale with couplings that are
predicted to be measurably non-universal within the KK tower. In addition, in
the case of flat compactifications, fermion localization can result in KK
graviton and gauge field flavor changing interactions.
