Regularization of the Kepler Problem on the Three-sphere In this paper we regularize the Kepler problem on $S^3$ in several different ways. First, we perform a Moser-type regularization. Then, we adapt the Ligon-Schaaf regularization to our problem. Finally, we show that the Moser regularization and the Ligon-Schaaf map we obtained can be understood as the composition of the corresponding maps for the Kepler problem in Euclidean space and the gnomonic transformation. Keywords:Kepler problem on the sphere, Ligon-Shaaf regularization, geodesic flow on the sphereCategory:70Fxx