The motion of drops suspended in another fluid has a wide variety of practical applications. These include the flow of oil and water through pipelines, the recovery of oil through porous rock and polymer processing. The lateral migration of deformable particles in shear flows has been the subject of many investigations. The motion of deformable drops suspended in a linear shear flow at non-zero Reynolds numbers is studied by numerical simulations in two dimensions. In this study the full Navier-Stokes equations are solved by a finite difference/front tracking method. A drop migrates to an equilibrium position in a horizontal shear flow. In this part, effects of non-dimensional parameters such as Particle Reynolds number, Gravity Reynolds number, density ratio, viscosity ratio, drop size and capillary number are investigated. It was found that a drop, which in the absence of shear would settle to the bottom of the channel, is actually lifted and obtains an equilibrium position away from the channel floor when the Reynolds number is large enough. It is expected that at very high shear rates, drops with high density ratios would suspend in the flow. As the capillary number, the Reynolds number, viscosity ratio or drop size decreases, the equilibrium position moves closer to bottom of channel. The drop velocity is observed to increase with increasing capillary number, viscosity ratio and Reynolds number. The drops are more deformed with increasing the capillary number. To validate the present calculations, some typical results are compared with the available predictions, which confirm that the present approach is reliable in predicting the drop migration. In second part, we investigated the effects of non-dimensional parameters such as bulk Reynolds number, gravity Reynolds number and density ratio on the motion of suspension of buoyant drops in a shear flow. It is observed that under some conditions, flow experiences solid-like or fluid-like behavior and the transition between these two behaviors are numerically investigated. As the bulk Reynolds number decreases or the gravity Reynolds number and density ratio increases, average equilibrium position of drops moves closer to bottom of channel. It is observed that average drops density distribution in upper half of channel increases by increasing the bulk Reynolds number or decreasing the gravity Reynolds number and density ratio. The fluctuation energy of the flow across the channel (suspension temperature) was also computed for different flow conditions. It is of interest because it is closely related to the diffusivity of suspension. The fluctuation energy is lowest near the channel floor where flow experiences solid-like behavior and the interaction between drops is weak. It increases as one moves away from the channel floor and becomes maximum in the region that flow experiences fluid-like behavior. It vanishes in the wall regions where no drops are present. The fluctuation energy decreases with Capillary number since the interaction between drops gets weaker as the Capillary number is raised. Keywords: Equilibrium position, Shear flow, Reynolds number, Gravity Reynolds number, Capillary number, Density ratio, Viscosity ratio, Suspension of buoyant drops, Fluctuation energy.