This thesis is divided into three main parts. In the first of these (comprising chapters 1 and 2)
we present the physical context of the research and cover the basic geometric background we
will need to use throughout the rest of this thesis.
In the second part (comprising chapters 3 to 5) we motivate and develop the strong homogeneity
theorem for supergravity backgrounds. We go on to prove it directly for a number of
top-dimensional Poincaré supergravities and furthermore demonstrate how it also generically
applies to dimensional reductions of those theories.
In the third part (comprising chapters 6 and 7) we show how further specialising to the case
of symmetric backgrounds allows us to compute complete classifications of such backgrounds.
We demonstrate this by classifying all symmetric type IIB supergravity backgrounds. Next we
apply an algorithm for computing the supersymmetry of symmetric backgrounds and use this to
classify all supersymmetric symmetric M-theory backgrounds.