Non-Euclidean geometries are types of geometry that differ from the traditional Euclidean geometry, which is based on the postulates of the ancient Greek mathematician Euclid. In non-Euclidean geometries, the parallel postulate is replaced, leading to different properties of shapes and spaces. Two main types are hyperbolic geometry, where parallel lines diverge, and elliptic geometry, where they converge. These geometries have important applications in various fields, including physics and cosmology. For example, Albert Einstein's theory of general relativity uses non-Euclidean geometry to describe the curvature of space-time around massive objects, fundamentally changing our understanding of gravity and the universe.