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Hyperbolic geometry (also called Lobachevskian geometry or
Bolyai–Lobachevskian geometry) is a non-Euclidean geometry.
The parallel postulate of Euclidean geometry is replaced with:

For any given line R and point P not on R,
in the plane containing both line R and
point P there are at least two distinct lines through P that do not intersect R.
(compare this with Playfair's axiom, the modern version of
Euclid's parallel postulate)

Hyperbolic plane geometry is also the geometry of saddle surfaces and pseudospherical surfaces, surfaces with a constant negative Gaussian curvature.

A modern use of hyperbolic geometry is in the theory of special relativity, particularly Minkowski spacetime and gyrovector space.