Klein, one of the great formative influences on the development of modern geometry, was born in Düsseldorf, Germany, and studied at Bonn, Göttingen, and Berlin. He worked with Sophus Lie – a collaboration that was particularly fruitful for both of them and led to the theory of groups of geometrical transformations. This work was later to play a crucial role in Klein's own ideas on geometry.

Klein took up the chair in mathematics at the University of Erlangen in 1872 and his inaugural lecture was the occasion of his formulation of his famous Erlangen Programm, a suggestion of a way in which the study of geometry could be both unified and generalized. Throughout the 19th century, with the work of such mathematicians as Karl Friedrich Gauss, Janós Bolyai, Nikolai Lobachevsky, and Bernhard Riemann, the idea of what a ‘geometry’ could be had been taken increasingly beyond the conception Euclid had of it and Klein's ideas helped show how these diverse geometries could all be seen as particular cases of one general concept. Klein's central idea was to think of a geometry as the theory of the invariants of a particular group of transformations. His Erlangen Programm was justly influential in guiding the further development of the subject. In particular Klein's ideas led to an even closer connection between geometry and algebra.

Klein also worked on projective geometry, which he generalized beyond three dimensions, and on the wider application of group theory, for example, to the rotational symmetries of regular solids. His name is remembered in topology for the Klein bottle, a one-sided closed surface, not constructible in three-dimensional Euclidean space. In 1886 Klein took up a chair at Göttingen and was influential in building Göttingen up into a great center for mathematics.