Numerical solutions of the equations describing flow of variably fluidized Coulomb mixtures predict key features of dry granular avalanches and water-saturated debris flows measured in physical experiments. These features include time-dependent speeds, depths, and widths of flows as well as the geometry of resulting deposits. Three-dimensional (3-D) boundary surfaces strongly influence flow dynamics because transverse shearing and cross-stream momentum transport occur where topography obstructs or redirects motion. Consequent energy dissipation can cause local deceleration and deposition, even on steep slopes. Velocities of surge fronts and other discontinuities that develop as flows cross 3-D terrain are predicted accurately by using a Riemann solution algorithm. The algorithm employs a gravity wave speed that accounts for different intensities of lateral stress transfer in regions of extending and compressing flow and in regions with different degrees of fluidization. Field observations and experiments indicate that flows in which fluid plays a significant role typically have high-friction margins with weaker interiors partly fluidized by pore pressure. Interaction of the strong perimeter and weak interior produces relatively steep-sided, flat-topped deposits. To simulate these effects, we compute pore pressure distributions using an advection-diffusion model with enhanced diffusivity near flow margins. Although challenges remain in evaluating pore pressure distributions in diverse geophysical flows, Riemann solutions of the depth-averaged 3-D Coulomb mixture equations provide a powerful tool for interpreting and predicting flow behavior. They provide a means of modeling debris flows, rock avalanches, pyroclastic flows, and related phenomena without invoking and calibrating rheological parameters that have questionable physical significance. ¿ 2001 American Geophysical Union