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.