**Abstract: **In this overview talk we investigate several problems in relativity and
particle physics where symmetries play a central role; in all cases
geometric properties of Lie groups and their quotients are related to
physical effects.
In the first part we discuss isometry groups of exact solutions in
general relativity, relating the algebraic properties of these groups to
physical properties of the spacetimes; we also generalise deformed
special relativity (DSR) by describing gravity as a gauge theory of the
de Sitter group, finding that Minkowski space has a connection with
torsion; after that we give a formulation of gravity as a topological
theory with added linear constraints that reduce the symmetries of the
topological theory to those of general relativity.
In the second part we study CP violation in the electroweak sector of
the standard model and certain extensions of it. We quantify fine-tuning
in the observed magnitude of CP violation by determining a natural
measure on the space of CKM matrices, comparing different possible
choices. The generic fine-tuning problem in the standard model is
removed by a measure that incorporates the observed quark masses, which
suggests a close relation between a mass hierarchy and suppression of CP
violation. Going beyond the standard model by adding a left-right
symmetry however spoils the result, making such additional symmetries
appear less natural.