A study is carried out for the flow of viscoelastic fluid contained between a stationary inner sphere and a rotating outer sphere. A sequence of flow transitions that is unique to viscoelastic fluids is observed in the experiments. It is found that a ＇traveling cell＇, with roll-cell-like characteristics, is generated in the polar region, and then propagates toward the equatorial region when a dimensionless parameter (a measure of strength of the elasticity against the shear viscosity) is increased. In order to investigate the structure and mechanism of the traveling cell, a numerical analysis is performed using a constitutive equation of the Giesekus model. Results of the numerical simulation revealed that the elasticity of the fluid strongly influences the flow in the polar region, where the inertia effect of the outer sphere rotation is minimal, destabilizing the flow forming a pair of weak vortices at the polar region. It was further shown that the pair travel toward the equatorial region in a different manner depending upon the speed of rotation.