A knowledge of the terminal falling velocity of particles in stationary and moving fluid streams is frequently needed in a wide spectrum of process engineering applications including liquid-solid separations, fluidization and transportation of solids, falling ball viscometry, drilling applications (Gavignet and Sobey, 1989; Li and Kuru, 2003), etc. The terminal falling velocity of a particle depends upon a rather large number of variables including the size, shape, and density of particles, its orientation, properties of the liquid medium (density, rheology), size and shape of the fall vessels, and whether the liquid is stationary or moving. The discussion presented in this chapter is mainly concerned with the motion of particles falling freely in quiescent fluids, albeit some of the results apply equally well when a particle is held stationary in a stream of moving fluids. Perhaps the time-independent fluid behavior represents the most commonly encountered type of fluid behavior. In this chapter, consideration is therefore given to the influence of fluid characteristics on global quantities such as drag coefficient and sedimentation velocity as well as on the detailed structure of the flow field for the steady motion of rigid spherical and nonspherical particles. Within the framework of the time-independent fluid behavior, attention is given to the particle motion in shear-thinning and shear-thickening liquids (without a yield stress) in this chapter and the analogous treatment for visco-plastic liquids is presented in Chapter 4. Likewise, the effect of confining boundaries on the hydrodynamic behavior of particles is considered in Chapter 10. A terse discussion of the significant results on particle motion in incompressible Newtonian fluids is also included here, not only because it is a special case of the time-independent fluid behavior, but it also lays the foundation for the subsequent treatment for non-Newtonian fluids. It is convenient to begin with the motion of a spherical particle in a Newtonian fluid medium.