This paper gives n-dimensional analogues of the Apollonian circle packings in parts I and II. Those papers considered circle packings described in terms of their Descartes configurations, which are sets of four mutually touching circles. They studied packings that had integrality properties in terms of the curvatures and centers of the circles. Here we consider collections of n-dimensional Descartes configurations, which consist of n + 2 mutually touching spheres.We work in the space M n D of all n-dimensional oriented Descartes configurations parametrized in a coordinate system, ACC-coordinates, as those (n + 2) × (n + 2) real matrices W with, and Q D,n and Q W,n are their corresponding symmetric matrices. On the parameter space M n D of ACC-matrices the group Aut(Q D,n ) acts on the left, and Aut(Q W,n ) acts on the right. Both these groups are isomorphic to the (n + 2)-dimensional Lorentz group O(n + 1, 1), and give two different "geometric" actions. The right action of Aut(Q W,n ) (essentially) corresponds to Möbius transformations acting on the underlying Euclidean space R n while the left action of Aut(Q D,n ) is defined only on the parameter space M n D .1 Partially supported by NSF grants DMS-0070574, DMS-0245526 and a Sloan Fellowship. This author is also affiliated with Dalian University of Technology, China.We introduce n-dimensional analogues of the Apollonian group, the dual Apollonian group and the super-Apollonian group. These are finitely generated groups in Aut(Q D,n ), 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 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 there exist rational Apollonian cluster ensembles (all curvatures rational) and strongly rational Apollonian sphere ensembles (all ACC-coordinates rational).