  This thesis presents a framework in which to explore kinematical symmetries
beyond the standard Lorentzian case. This framework consists of an algebraic
classification, a geometric classification, and a derivation of the geometric
properties required to define physical theories on the classified spacetime
geometries. The work completed in substantiating this framework for
kinematical, super-kinematical, and super-Bargmann symmetries constitutes the
body of this thesis.
  To this end, the classification of kinematical Lie algebras in spatial
dimension $D = 3$, as presented in [3, 4], is reviewed; as is the
classification of spatially-isotropic homogeneous spacetimes of [5]. The
derivation of geometric properties such as the non-compactness of boosts,
soldering forms and vielbeins, and the space of invariant affine connections is
then presented.
  We move on to classify the $\mathcal{N}=1$ kinematical Lie superalgebras in
three spatial dimensions, finding 43 isomorphism classes of Lie superalgebras.
Once these algebras are determined, we classify the corresponding
simply-connected homogeneous (4|4)-dimensional superspaces and show how the
resulting 27 homogeneous superspaces may be related to one another via
geometric limits.
  Finally, we turn our attention to generalised Bargmann superalgebras. In the
present work, these will be the $\mathcal{N}=1$ and $\mathcal{N}=2$
super-extensions of the Bargmann and Newton-Hooke algebras, as well as the
centrally-extended static kinematical Lie algebra, of which the former three
all arise as deformations. Focussing solely on three spatial dimensions, we
find $9$ isomorphism classes in the $\mathcal{N}=1$ case, and we identify $22$
branches of superalgebras in the $\mathcal{N}=2$ case.
