‘Geometry’ begins with the Euclidean geometry, the geometry with which most people are familiar. Euclid's Elements, written in around 300bc, started with five axioms from which Euclid deduced a large body of geometrical theorems. The parallel postulate axiom is explained in greater detail as it is more complicated than the other axioms, and involves the infinite in a fundamental way. Spherical geometry and hyperbolic geometry (using the disc model) are described to provide more information on the parallel postulate. Finally, the curvature of space and manifolds, the study of which is one of the most important branches of mathematics, are discussed.