Hyperbolic geometry, a branch of non-Euclidean geometry characterized by constant negative curvature, has profoundly influenced both pure mathematics and theoretical physics. Its rejection of Euclid’s parallel postulate leads to a universe where the sum of the angles of a triangle is less than 180 degrees, and distances grow exponentially from a point. In recent decades, hyperbolic geometry has found critical applications in cosmology, providing models for the large-scale structure of the universe, the behavior of spacetime in relativity, and the mathematics underlying cosmic inflation and quantum gravity. This article surveys the foundational concepts of hyperbolic geometry, explores its mathematical models, and delves into its pivotal role in modern cosmological theories and observations.