Abstract Geometry growth laws for morphological change are developed and examined for the class of dynamic problems where surface diffusion is the only transport mechanism and hence volume is conserved, attachment kinetics is treated, and the only driving force for surface motion is the reduction in total surface free energy. The two limiting laws in the isotropic case are: motion by the Laplacian of mean curvature as originally derived by Mullins, and motion by the difference between mean curvature and the average of mean curvature. A general law linking these limiting laws is formulated, and derived both from a physical model and from gradient flows. Anisotropic laws are given. We survey possible mathematical techniques for studying interface motion under all these laws. Among these are possible applications of modified phase field methods.