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This paper discusses the problem of approximating data points in n-dimensional Euclidean space using spherical and ellipsoidal surfaces. A closed form solution is provided for spherical approximation, while an efficient, globally optimal solution for the ellipsoidal problem is proposed in terms of semidefinite programming. In addition, the paper presents a result for robust fitting in presence of outliers, and illustrates the theory with several numerical examples. A brief survey is also presented on the solutions to other relevant geometric approximation problems, such as ellipsoidal covering of convex hulls and pattern separation.