In this dissertation, we investigate via numerical computations the dynamicsof elastic capsules (made from a thin strain-hardening elastic membrane) in two microfluidic channels of cross-junction and T-junction geometries. For the cross-junction microfluidic channel, we consider an initially spherical capsule with a size smaller than the cross-section of the square channels comprising the cross-junction, and investigate the effects of the capsule size, flow rate, and lateral flow rates on the transient dynamics and deformation of low-viscosity and equiviscous capsules. In addition, we also study the effects of viscosity ratio on the transient capsule dynamics and deformation. Our investigation shows that the intersecting lateral flows at the cross-junction act like a constriction. Larger capsules, higher flow rates and higher intersecting lateral flows result in stronger hydrodynamic forces that cause a significant capsule deformation, i.e., the capsule’s length increases while its height decreases significantly. The capsule obtains different dynamic shape transitions due to the asymmetric shape of the cross-junction. Larger capsules take more time to pass through the cross-junction owning to the higher flow blocking. As the viscosity ratio decreases, the capsule’s transient deformation increases and tail formation develops transiently, especially for low-viscosity capsules owing to the normal-stress effects of the surrounding fluid on the capsule’s interface. However, the viscosity ratio does not affect much the capsule velocity due to a weak inner circulation. Our findings suggest that the tail formation of low-viscosity capsule may promote membrane breaking and thus drug release of pharmaceutical capsules in the microcirculation. Furthermore, we investigate via numerical computations the motion of an elastic capsule (made from an elastic membrane obeying the strain-hardening Skalak law) flowing inside a microfluidic T-junction device. In particular, we consider the effects of the capsule size, flow rate, lateral flow rate, and fluid viscosity ratio on the motion of the capsule in the T-junction micro-channel. As the capsule’s initial lateral position increases, the capsule moves faster and reaches different final lateral positions. As the capsule size increases, the gap between the capsule’s surface and the channel wall decreases. This results in the development of stronger hydrodynamic forces and a decrease in the capsule velocity due to flow blocking. As the capsule size increases, there is a small lateral migration towards the micro-channel centerline, which is the low-shear region of the T-junction micro-channel. This migration is in agreement with experimental and numerical studies on non-inertial lateral migration of vesicles in bounded Poiseuille flow by Coupier et al. [13] who showed that the combined effects of the walls and of the curvature of the velocity profile induce a lateral migration toward the centerline of the channel. As the capillary number Ca increases, the stronger hydrodynamic forces cause the capsule to extend along the flow direction (i.e., the capsule’s length Lx increases as the capsule enters the T-junctions and decreases as the capsule exits the T-junction). There is a small lateral migration away from the micro-channel centerline as the flow rate Ca increases. The capsule lateral position zc, main-flow velocity Ux and migration velocity Uz are practically not affected by the fluids viscosity ratio λ. As the channel’s lateral flow rate increases, the capsule migrates downwards towards the bottom of the device. Our findings on the lateral migration in the T-junction micro-channel suggest that there is a great potential for designing a T-junction microfluidic device that can be used to manipulate artificial and biological capsules.