An account, as non-technical as possible, is given of the invariance properties that have important applications in elementary particle physics. These can be classified in several ways. Firstly some are related to continuous groups (rotations), others to discrete groups (inversions). Secondly some are related to groups of well-known geometrical or kinematical significance (ordinary space time), others are not and are peculiar to elementary particle physics (isobaric space). Finally some are, it appears, rigorously valid while others are of approximate nature. The discussion aims to cover all cases and their interpretation. In as far as possible explicit use of quantum field theory techniques is avoided.