The standard diffusion-convection theory of cosmic-ray transport is unable to properly describe solar modulation at the highest rigidities where the gyroradius and mean free path of particles become large and the diffusion picture breaks down. Present models need to invoke an ad hoc upper cutoff rigidity. We report on an attempt to extend the theoretical description of solar modulation to high rigidities, including the regime where diffusion approximation is not applicable. A scheme retaining the full directional distribution is outlined that includes scattering but does not rely upon the concept of diffusive streaming and thus remains valid at arbitrarily large values of the scattering mean free path. The scattering process is briefly discussed. We consider isotropic scattering and the simplest form of anisotropic scattering, but our formalism can accommodate more complex scattering processes as well. The application of the method is illustrated by numerical simulations obtained in a simplified two-dimensional scenario which includes a flat heliospheric current sheet (HCS). We find that the anisotropies predicted in the inner heliosphere turn out to be largely independent of the conditions assumed at the outer boundary, and the anisotropies decrease gradually as the particle rigidity increases. Isotropic scattering gives rise to very small anisotropies in the inner heliosphere, where the solutions converge to an isotropic three-dimensional (3-D) force-field solution. Anisotropic scattering, however, can produce reasonable anisotropies at the Earth's orbit. While the model is still greatly idealized and we do not attempt quantitative fits to observations at this stage, some important tendencies can already clearly be demonstrated. For instance, the polarity dependence of the solar diurnal variation is reproduced, the sense of the observed phase shift of the solar diurnal wave at polarity reversals of the sun is correctly predicted. The present two-dimensional (2-D) model fails to explain the sector-dependent North-South anisotropy, which is a direct result of the wavy structure of the current sheet and cannot be interpreted in terms of a 2-D model. ¿ 1999 American Geophysical Union