Non-Euclidean geometry refers to any type of geometry that differs from the traditional Euclidean geometry established by the ancient Greek mathematician Euclid. In Euclidean geometry, the familiar rules apply, such as the sum of angles in a triangle equaling 180 degrees. Non-Euclidean geometries, however, explore different properties, such as curved surfaces, where these rules do not hold. There are two main types of non-Euclidean geometry: hyperbolic and elliptic. In hyperbolic geometry, the sum of angles in a triangle is less than 180 degrees, while in elliptic geometry, it is greater than 180 degrees. These concepts have important applications in fields like physics and cosmology.