  This paper gives $n$-dimensional analogues of the Apollonian circle packings
in parts I and II. We work in the space $\sM_{\dd}^n$ of all $n$-dimensional
oriented Descartes configurations parametrized in a coordinate system,
ACC-coordinates, as those $(n+2) \times (n+2)$ real matrices $\bW$ with $\bW^T
\bQ_{D,n} \bW = \bQ_{W,n}$ where $Q_{D,n} = x_1^2 +... + x_{n+2}^2 -
\frac{1}{n}(x_1 +... + x_{n+2})^2$ is the $n$-dimensional Descartes quadratic
form, $Q_{W,n} = -8x_1x_2 + 2x_3^2 + ... + 2x_{n+2}^2$, and $\bQ_{D,n}$ and
$\bQ_{W,n}$ are their corresponding symmetric matrices. There are natural
actions on the parameter space $\sM_{\dd}^n$. We introduce $n$-dimensional
analogues of the Apollonian group, the dual Apollonian group and the
super-Apollonian group. These are finitely generated groups with the following
integrality properties: the dual Apollonian group consists of integral matrices
in all dimensions, while the other two consist of rational matrices, with
denominators having prime divisors drawn from a finite set $S$ depending on the
dimension. We show that the the Apollonian group and the dual Apollonian group
are finitely presented, and are Coxeter groups. We define an Apollonian cluster
ensemble to be any orbit under the Apollonian group, with similar notions for
the other two groups. We determine in which dimensions one can find rational
Apollonian cluster ensembles (all curvatures rational) and strongly rational
Apollonian sphere ensembles (all ACC-coordinates rational).
