Euclidean space. This fact has several implications. First of all, one can consider harmonic functions which transform under irreducible representations of the conformal group or of its subgroups (such as the group of rotations or translations). Proceeding in this fashion, one systematically obtains the solutions of the Laplace equation which arise from separation of variables such as spherical harmonic solutions and Fourier series. By taking linear superpositions of these solutions, one can produce large classes of harmonic functions which can be shown to be dense in the space of all harmonic functions under suitable topologies. Second, one can use conformal symmetry to understand such classical tricks and techniques for generating harmonic functions as the Kelvin transform and the method of images. Third, one can use conformal transforms to map harmonic functions in one domain to harmonic functions in another domain. The most common instance of such a construction is to relate harmonic functions on a disk to harmonic functions on a half-plane. Fourth, one can use conformal symmetry to extend harmonic functions to harmonic functions on conformally flat Riemannian manifolds. Perhaps the simplest such extension is to consider a harmonic function defined on the whole of Rn (with the possible exception of a discrete set of singular points) as a harmonic function on the -dimensional sphere. More complicated situations can also happen. For instance, one can obtain a higher-dimensional analog of Riemann surface theory by expressing a multi-valued harmonic function as a single-valued function on a branched cover of Rn or one can regard harmonic functions which are invariant under a discrete subgroup of the conformal group as functions on a multiply connected manifold or orbifold. Two dimensions From the fact that the group of conformal transforms is infinite-dimensional in two dimensions and finite-dimensional for more than two dimensions, one can surmise that potential theory in two dimensions is different from potential theory in other dimensions. This is correct and, in fact, when one realizes that any two-dimensional harmonic function is the real part of a complex analytic function, one sees that the subject of two-dimensional potential theory is substantially the same as that of complex analysis. For this reason, when speaking of potential theory, one focuses attention on theorems which hold in three or more dimensions. In this connection, a surprising fact is that many results and concepts originally discovered in complex analysis (such as Schwarz's theorem, Morera's theorem,