The pulsatile blood flows in solid blood vessels are investigated numerically in order to understand some physiological phenomena in arteries. For the geometry of the blood vessels, one-point stenosed and periodically stenosed blood vessels are considered. Taking advantage of axisymmetry in the problem, the stream function-vorticity formulation is used for the governing equations of the fluid flows. All the computations are performed by using the ADI scheme of the finite difference method on the numerically generated boundary-fitted orthogonal curvilinear coordinate systems. The flow fields are found to be dramatically different depending on the Strouhal number. When the Strouhal number is O(1) or larger, the flow field is quite dynamic in the sense that the vortices formed during the previous period survive and exert residual stress on the blood vessel wall. On the other hand, when the Strouhal number is as small as O(10^-2), the flow fields are found to be in the quasi-steady state. The computation results suggest that the deterioration of endothelial cells may occur due to strong local flow fields near the stenosis and that the probability of platelet attachment to the blood vessel wall is higher in the region behind the stenosis. From the results for the periodically stenosed vessel, the so-called steady streaming phenomenon is confirmed. The steady streaming effect in a wavy channel is expected to increase the heat and mass transfer rate without making the flow turbulent.