Complex fluids, especially fluids containing polymer molecules, are frequently encountered in everyday life from foods, paints, to cosmetics. Polymeric fluids are usually viscoelastic and do not flow like water. As a result, these fluids can exhibit flow instabilities even at low Reynolds number (Re) where viscous forces dominate inertial forces, and a new type of turbulence - purely elastic turbulence. It has been demonstrated that these nonlinear behaviors are arised from elastic stresses and have been observed in flows with curved streamlines. The first part of this work investigates flow instabilities of viscoelastic fluids in microchannel system. Experiments are performed in a long, straight microchannel where the flow is perturbed by placing a variable number of cylinders (0 ≤ n ≤ 15). Results show that the initial disturbance is sustained, in the form of temporal velocity fluctuations, far downstream from the obstacles (200× channel width) in the parallel shear geometry above certain Weissenberg number (Wi). These temporal fluctuations in velocity increase nonlinearly with Wi. Above a critical Wi (Wi > 5.4) and a critical number of obstacles (n ≥ 2), a sharply increase of velocity fluctuations together with a hysteresis loop indicate the presence of a subcritical elastic instability. It is also observed that the initial disturbances can be spreaded far upstream and the upstream velocity fluctuations increase linearly with Wi suggesting the existence of a linear instability upstream. The second part of this thesis is concerned with the rheological characterization of complex fluids in high-shear-rate environments. Such environments are found in lubrication and coating processes as well as in flow through porous media. Microfluidic technology is used due to its small length scale so that the flow remains in the low Re regime (Re << 1) while attaining high shear-rates (up to 104 s-1). Measurements of viscosity of complex fluids including polymeric solutions and human blood plasma at high shear-rates are performed using microfluidic-based rheometry. Viscosity is estimated by measuring the pressure drop along a microchannel using in-situ pressure sensors. Finally, a novel method is proposed and implemented to measure relaxation times of viscoelastic fluids at low strain.