  Kendall's Shape Theory covers shapes formed by $N$ points in $\mathbb{R}^d$
upon quotienting out the similarity transformations. This theory is based on
the geometry and topology of the corresponding configuration space: shape
space. Kendall studied this to build a widely useful Shape Statistics
thereupon. The corresponding Shape-and-Scale Theory -- quotienting out the
Euclidean transformations -- is useful in Classical Dynamics and Molecular
Physics, as well as for the relational `Leibnizian' side of the Absolute versus
Relational Motion Debate. Kendall's shape spaces moreover recur withing this
`Leibnizian' Shape-and-Scale Theory. There has recently been a large expansion
in diversity of Kendall-type Shape(-and-Scale) Theories. The current article
outlines this variety, and furthermore roots it in solving the poset of
generalized Killing equations. This moreover also places a first great bound on
how many more Shape(-and-Scale) Theories there can be. For it is nongeneric for
geometrically-equipped manifolds -- replacements for Kendall's $\mathbb{R}^d$
carrier space (absolute space to physicists) - to possess any generalized
Killing vectors. Article II places a second great bound, now at the topological
level and in terms of which Shape(-and-Scale) Theories are technically
tractable. Finally Article III explains how the diversity of Shape(-and-Scale)
Theories - from varying which carrier space and quotiented-out geometrical
automorphism group are in use - constitutes a theory of Comparative Background
Independence: a topic of fundamental interest in Dynamics, Gravitation and
Theoretical Physics more generally. Article I and II's great bounds moreover
have significant consequences for Comparative Background Independence.
