The author surveys recent developments in geometry and argues that a unifying principle is the study of groups of spatial transformations, which determine which properties are geometric. He traces how projective methods have been extended to encompass metric relations via a fundamental conic, and catalogs alternative approaches such as line geometry, inverse-radius transformations, rational mappings, and topology. The essay examines synthetic versus analytic methods, the use of manifolds of arbitrary dimension, invariant-theoretic and constant-curvature treatments, and the relation between specialized disciplines. Concluding notes clarify technical points and distinguish the abstract mathematical viewpoint from related perspectives.