In this study, several important properties of inertial bubble and solid particle suspensions have been characterized using numerical simulations. These properties include the hindered settling velocity and microstructure in solid particle suspensions, the scaling of the velocity variance and hydrodynamic diffusivity in solid particle suspensions, the hindered rise velocity and microstructure in suspensions of monodisperse, spherical, non-coalescing bubbles, and the lift force on bubbles in a sheared suspension. In order to simulate suspensions of monodisperse, spherical, non-coalescing bubbles, a lattice-Boltzmann boundary rule was designed. On the last subject, the lift force in a sheared suspension, experiments have also been conducted to verify the simulations.