Non-Euclidean geometry is a type of geometry that differs from the traditional Euclidean geometry established by the ancient Greek mathematician Euclid. In Euclidean geometry, the parallel postulate states that through a point not on a line, there is exactly one parallel line. Non-Euclidean geometry challenges this idea, allowing for multiple parallel lines or none at all. There are two main types of non-Euclidean geometry: hyperbolic geometry and elliptic geometry. In hyperbolic geometry, the space is curved outward, leading to unique properties, such as the sum of angles in a triangle being less than 180 degrees. In elliptic geometry, the space is curved inward, resulting in no parallel lines and the sum of angles in a triangle exceeding 180 degrees.