To contribute to the understanding of flow phenomena in abdominal aortic aneurysms, numerical computations of pulsatile flows through aneurysm models and a stability analysis of these flows were carried out. The volume flow rate waveforms into the aneurysms were based on measurements of these waveforms, under rest and exercise conditions, of patients suffering abdominal aortic aneurysms. The Reynolds number and Womersley number, the dimensionless quantities that characterize the flow, were varied within the physiologically relevant range, and the two geometric quantities that characterize the model aneurysm were varied to assess the influence of the length and maximal diameter of an aneurysm on the details of the flow. The computed flow phenomena and the induced wall shear stress distributions agree well with what was found in PIV measurements by Salsac et al. (J. Fluid Mech., vol. 560, 2006, pp. 19–51). The results suggest that long aneurysms are less pathological than short ones, and that patients with an abdominal aortic aneurysm are better to avoid physical exercise. The pulsatile flows were found to be unstable to three-dimensional disturbances if the aneurysm was sufficiently localized or had a sufficiently large maximal diameter, even for flow conditions during rest. The abdominal aortic aneurysm can be viewed as acting like a 'wavemaker' that induces disturbed flow conditions in healthy segments of the arterial system far downstream of the aneurysm; this may be related to the fact that one-fifth of the larger abdominal aortic aneurysms are found to extend into the common iliac arteries. Finally, we report a remarkable sensitivity of the wall shear stress distribution and the growth rate of three-dimensional disturbances to small details of the aneurysm geometry near the proximal end. These findings suggest that a sensitivity analysis is appropriate when a patient-specific computational study is carried out to obtain a quantitative description of the wall shear stress distribution.