In October 1872, at just 23 years old and recently appointed professor at the University of Erlangen, Felix Klein presents in an inaugural lecture a research program that would introduce a radically new conceptual order into a field that until then had been fragmented into apparently unrelated geometries: Euclidean, projective, affine, and the hyperbolic and elliptic non-Euclidean geometries discovered decades earlier by Lobachevsky, Bolyai, and Riemann. Klein's central idea, known since then as the Erlangen Program, is dazzlingly simple in its formulation but of immense unifying scope: each geometry can be completely characterized, not by its particular axioms, but by the group of transformations under which its properties remain invariant. Euclidean geometry is the study of properties preserved under the group of rigid motions (rotations, translations, reflections); projective geometry, under the broader group of projective transformations, which preserves neither distances nor angles but does preserve incidence relations; and so on for each known geometry, organized into a clear hierarchy according to the inclusion of some transformation groups within others. Klein develops this vision in close collaboration with the Norwegian mathematician Sophus Lie, with whom he shares an interest in linking geometry and group theory from the late 1860s onward. The program not only provided an elegant classification of geometry as known at the time, but also established a methodological principle — understanding a mathematical object through its symmetry group — that would become one of the most fertile organizing ideas of 20th-century mathematics and physics, from general relativity to particle physics and crystallography, and that a century later would explicitly inspire mathematicians such as Jacques Tits in his theory of buildings and Mikhail Gromov in his geometric treatment of infinite groups.