The following problem is solved: What are all the "different" separable coordinate systems for the Laplace-Beltrami eigenvalue equation on the n-sphere Sn and Euclidean n-space Rn and how are they constructed? This is achieved through a combination of differential geometric and group theoretic methods. A graphical procedure for construction of these systems is developed that generalizes Vilenkin's construction of polyspherical coordinates. The significance of these results for exactly soluble dynamical systems on these manifolds is pointed out. The results are also of importance for the analysis of the special functions appearing in the separable solutions of the Laplace-Beltrami eigenvalue equation on these manifolds.