This thesis focuses on numerical and experimental investigations of shear-thinning viscoelastic (VE) fluid flow in confined geometries, such as porous media and microchannels for applications in polymer processing, heat transfer devices, biological and biomedical systems, and enhanced oil recovery. The numerical research focuses on the hydrodynamic performance of a VE fluid with particular attention to vortex patterns that develop downstream of smooth contractions/expansions. It was found that there are critical regions, in which the behavior of VE fluid flow is subjected to a change through different pore geometries. The experimental work focuses on the retention phenomenon of hydrolyzed polyacrylamide polymers, using microfluidic chips in which the effect of single pore size on retention is measured. From these experiments, the contribution of different retention mechanisms was reported through different sizes of microchannel. It was found that polymer orientation plays a role in the measured absolute and relative adsorption in different concentrations.