Let © be a finite group of transformations of three-dimensional Euclidean space, such that the distance between any two points is preserved by all transformations of the group. Such a group is a group of orthogonal linear transformations of three variables, or, geometrically speaking, a group of rotations and rotatory inversions. Thirty-two groups of this type are important in crystallography and are known as the crystallographic classes. A function is said to have the symmetry of a given group if it remains invariant under all transformations of the group. Our problem is to determine all spherical harmonics of a given degree m and a given symmetry. It is sufficient to find a basis of these harmonics for all m and for all groups ©. Section I of this paper enumerates and classifies all groups of the desired type. In §11 we find the number of elements in a basis of all homogeneous polynomials of a given degree which have a given symmetry, applying a theorem of Molien. In §111 we find the number of elements in a basis of all spherical harmonics of a given degree which have a given symmetry. This is accomplished by associating with each group a generating function. In §IV we solve the problem proposed, using the results of §111. The required basis is found in terms of partial derivatives of 1/r, r denoting the distance from the origin. For certain simpler symmetries the basis is also expressed in terms of the associated Legendre functions. A particular case of this problem arose and was solved in another research problem, the aim of which was to compute approximately the electrostatic capacity of the cube (12, pp. 76-78). In generalizing this particular case, we were led to our results which were announced, without proof, in two notes (10; 11). Work in this problem has been done previously by Poole (13), Laporte (7), Bethe (1), Ehlert (4), and Hodgkinson (6), and recently by Stiefel (15). For the geometrical and algebraic background see Molien (9) and the bibliographies in Coxeter (3a) and Speiser (14). The present paper differs in two respects from preceding work on the subject. First, all groups are treated in a uniform manner, whereas previous papers are restricted to certain groups. Second, the generating function of §111 enables