Velocity addition of classical velocities possesses the mathematical structure of vector spaces which, in turn, form the setting for Euclidean geometry. In contrast, Einstein velocity addition of relativistic velocities seems to be structureless since it is neither commutative nor associative. However, I will demonstrate that Einstein velocity addition has a rich mathematical structure, giving rise to the new concept of hyperbolic vector spaces, called gyrovector spaces. The latter, in turn, form the setting for the familiar hyperbolic geometry of Bolyai and Lobachevsky just as vector spaces form the setting for Euclidean geometry. The talk is based on my two books listed below* and is accessible to students who are familiar with the basic elements of the vector space approach to Euclidean geometry. Undergraduate students are, therefore, welcome.

*A. A. Ungar, Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces (Dordrecht: Kluwer Acad., 2001).
A. A. Ungar, Analytic Hyperbolic Geometry: Mathematical Foundations and Applications (NJ: World Scientific, 2005).