The classical theory of spherical harmonics and the related families of orthogonal polynomials (Legendre, Chebyshev, and Gegenbauer polynomials) goes back to the eighteenth century and is one of the most beautiful subjects in mathematical physics and pure mathematics, with many applications. The talk will describe a far-reaching generalization, developed over the course of the last years in collaboration with Tomoyoshi Ibukiyama, in which the spherical harmonics are replaced by functions of n variables in a d-dimensional Euclidean space that are harmonic and homogeneous in each variable separately and are invariant under the diagonal action of the group O(d). (The classical theory corresponds to the case n=2.) This definition was originally motivated by an application to the theory of Siegel modular forms but turns out to lead to a very rich theory that is of interest in its own right, but that is also much more complicated than the classical one because the spaces of polynomials of fixed degree are now no longer one-dimensional. There is also a notion of "higher spherical functions", which are related to the higher spherical polynomials in the same way as Legendre functions of the second kind are related to Legendre polynomials, i.e., they are the non-polynomial solutions of the same systems of differential equations. If time permits, I will also discuss this and its relation to the theory of holonomic systems.