A procedure for the parametrization and description of the surface of a simply connected 3-D object is presented. The object does not need to be star-shaped. Starting from segmented volume data, a relational data structure describing the adjacency of local surface elements is generated. It is used to parametrize the surface, i.e., to define a continuous, one-to-one mapping from the surface of the original object to the surface of a unit sphere. As any simply connected object is topologically equivalent to a sphere, such a mapping always exists. The mapping is constrained by two requirements, minimization of distortions and preservation of area (up to a scaling factor). The former is formulated as the goal function of a nonlinear optimization problem and the latter as its constraints. This parametrization allows the systematical scanning of the object surface by the variation of two parameters. It enables us to represent the object surface as a series of spherical harmonic functions. The numerical coefficients in this series form an object-centered, surface-oriented descriptor of the object's form. Rotating the coefficients in parameter space and object space puts the object into a standard position and yields a spherical harmonic descriptor which is invariant to translations, rotations and (if desired) scaling of the object. This approach produces an extension of the elliptical Fourier descriptors for 2-D closed curves by Persoon & Fu, and Kuhl & Giardina. Potential applications are shape recognition, classification and differentiation, e.g., of anatomical objects in medical image analysis.