Hyperbolic geometry describes spaces with constant negative curvature, where distances grow exponentially and parallel lines diverge. Unlike Euclidean geometry, it naturally represents hierarchical and tree-like structures. In probability, hyperbolic spaces are used to study random walks, percolation, and mixing on trees and networks, capturing rapid expansion and boundary behavior. In machine learning, hyperbolic embeddings efficiently represent hierarchies in data such as knowledge graphs, language taxonomies, and social networks with far fewer dimensions than Euclidean models. In real life, hyperbolic geometry appears in internet routing, complex networks, neuroscience, and cosmology, where underlying structures exhibit exponential growth and hierarchical organization. Its ability to model large-scale structure with local simplicity makes it a powerful tool for understanding complex systems.