Geometry has been historically one of the first branches
of Mathematics to develop. Of fundamental
and lasting importance was Euclide's "Elements",

For centuries, Euclidean geometry was considered as
the only conceivable Geometry. At the turn of the
XIXth century, the existence of other consistent geometries
emerged as a scientific possibility through the discovery of
the concept of curvature by Carl-Friedrich Gauss and works of
Joseph-Louis de Lagrange, Nicolas Lobatchewski and
Janos Bolyai .

The broadening of the concept of Geometry continued
all along the XIXth century with fundamental contributions
by Bernhard Riemann and several other mathematicians.

At the turn of the XXth century, Henri Poincaré insisted
on the arbitrariness of definitions in Geometry in terms of their
relation to the space we live in. Nevertheless
evidence of the relevance of geometric concepts in many areas of
science in particular in modern Physics, continued to appear
all along the century. This is even the case for other areas
of Mathematics as the recent proof by Grigory Perelman
of a famous purely topological conjecture by Henri Poincaré exemplifies.