We present a gravitationally consistent method for calculating the ''kernel'' functions which relate global geophysical observables, such as the geoid, surface plate motions and core-mantle boundary (CMB) topography, to internal density heterogeneities in a viscous, compressible mantle possessing an arbitrary radial variation of viscosity. We show that the influence of finite mantle compressibility is substantial in the case of the predicted nonhydrostatic geoid, with the largest effects occurring at spherical harmonic degree l=2. On the basis of the geoid data we find that our best two-layer viscosity model possesses a factor of 9 jump in viscosity at 1200 km depth. We argue, however, that the inferences of viscosity obtained from the geoid data are sensitive to the model of internal density heterogeneity which is employed and that this sensitivity probably explains the differences between or own viscosity inferences and those obtained by others. Our viscous flow models possess a spherically symmetric viscosity distribution and they cannot therefore ''explain'' the toroidal component of the flow of the tectonic plates. We therefore present a new scheme for predicting plate motions based on an explicit coupling of poloidal and toroidal flows which we derive on the basis of the assumption that the individual plates are perfectly rigid. We show how this new description of surface plate motions induced by buoyancy forces in the mantle may be used to constrain the absolute value of the long-term (i.e., steady state) mantle viscosity. ¿American Geophysical Union 1991