The authors consider a generic configuration of regions, consisting of a collection of distinct compact regions ${ Omega_i}$ in $mathbb{R}^{n+1}$ which may be either regions with smooth boundaries disjoint from the others or regions which meet on their piecewise smooth boundaries $mathcal{B}_i$ in a generic way. They introduce a skeletal linking structure for the collection of regions which simultaneously captures the regions' individual shapes and geometric properties as well as the ``positional geometry'' of the collection. The linking structure extends in a minimal way the individual ``skeletal structures'' on each of the regions. This allows the authors to significantly extend the mathematical methods introduced for single regions to the configuration of regions.