The biomechanical interaction of stents and the arteries into which they are deployed is a potentially important consideration for long-term success. Adverse arterial reactions to excessive stress and the resulting damage have been linked to the development of restenosis. Complex geometric features often encountered in these procedures can confound treatment. In some cases, it is desirable to deploy a stent across a region in which the diameter decreases significantly over the length of the stent. This study aimed to assess the final arterial diameter and circumferential stress in tapered arteries into which two different stents were deployed (one stiff and one less stiff). The artery wall was assumed to be made of a strain stiffening material subjected to large deformations, with a 10% decrease in diameter over the length of the stent. A commercially available finite element code was employed to solve the contact problem between the two elastic bodies. The stiffer stent dominated over arterial taper, resulting in a nearly constant final diameter along the length of the stent, and very high stresses, particularly at the distal end. The less stiff stent followed more closely the tapered contour of the artery, resulting in lower artery wall stresses. More compliant stents should be considered for tapered artery applications, perhaps even to the exclusion of tapered stents.