This paper describes a technology for manipulating the shape of an implicitly modeled surface so that it follows a geometric flow. The proposed technology is a generalization of the method of level sets, which specifies how to manipulate the isosurfaces of greyscale functions, sampled on discrete grids. The proposed formulation projects the geometric motion of the level sets of a greyscale function onto its parametric representation. The resulting method provides a set of new capabilities for implicit models, including the computation of geometric flows, such as motion by mean curvature and the fitting of implicitly represented surfaces to volumetric data. This paper presents results for the computation of mean-curvature flow on the level sets of polynomials and the fitting of radial basis functions, arranged as dipoles, in an adaptive-refinement scheme to achieve surface fitting with variational implicits.