This article describes how the author carries out an investigation into the geometry of the three possible curvatures of the universe. The author begins the investigation by looking on the web and in books. She found that the general consensus was that there were three different possible curvatures of the universe, namely: (1) flat; (2) positive; or (3) negative. Also mentioned were intrinsic and extrinsic curvature. The author begins this article by considering the extrinsic and intrinsic curvature. The author discusses spherical geometry as the geometry that occurs on a surface which has intrinsic curvature such as a sphere and provides a brief history of non-Euclidean geometry. (Contains 13 figures.)