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We review the $N$-Body Problem in arbitrary dimension $d$ at the kinematicallevel, with modelling Background Independence in mind. In particular, we give astructural analysis of its reduced configuration spaces, decomposing thissubject matter into basic Topology, Geometry, Group Theory, Linear Algebra,Graph Theory and Order Theory. At the metric level, these configuration spacesof shapes form basic geometric series for 1- and 2-$d$: $\mathbb{S}^{N - 2}$and $\mathbb{CP}^{N - 2}$, though there are no more such series for $d \geq 3$.$d \geq 3$ also sees an onset of stratification. Casson's diagonal, for which$N = d + 1$, plays a critical role which we explain in simple Linear Algebraterms; these are moreover topologically spheres. $N = d$ and $N = d + 2$ havefurther significance as well, the latter as regards a counting notion ofgenericity of the isotropy groups and kinematical orbits realized. Theseobservations twin the $N$ = (3, 4, 5) progression in qualitative complexitythat almost all $N$-Body Problem work for concrete $N$ concentrates on with themuch better-known $N$ = (2, 3, 4) progression in 2-$d$: from intervals totriangles to quadrilaterals: a large source of intuitions and mathematicalanalogies. We furthermore provide an Order-Theoretic genericity criterion,which is almost always bounded by the double-slope $N = 2 d + 1$ line, thoughan accidental relation pushes $N$ up by 1 to 8 in 3-$d$. We finally considerrubber shapes, for which the configuration spaces are graphs: much simpler thanstratified manifolds, and yet containing quite a few of metric-level shapes'qualitative features. This singles out $N = 5$ and 6 in 1-$d$ and $N = 6$ and 8in $\geq$ 2-$d$ for the onsets of various graph-theoretical nontrivialities.

$N$-Body Problem: Minimal $N$'s for Qualitative Nontrivialities