Vascular damage induced by acute hypertension is preceded by a peculiar pattern where blood vessels show alternating regions of constrictions and dilations ("sausages-on-a-string"). The pattern occurs in the smaller blood vessels, and it plays a central role in causing the vascular damage. A related vascular pattern has been observed in larger vessels from several organs during angiography. In the larger vessels the occurrence of the pattern does not appear to be related to acute hypertension. A unifying feature between the phenomenon in large and small vessels seems to be an increase in vascular wall tension. Despite much research, the mechanisms underlying the "sausage" pattern have remained unknown. Here we present an anisotropic model of the vessel wall, and show that the "sausage" pattern can arise due to an instability of the vessel wall. The model reproduces many of the key features observed experimentally. Most importantly, it suggests that the "sausaging" phenomenon is neither caused by a mechanical failure of the vessel wall due to a high blood pressure, nor is it due to standing pressure waves caused by the beating of the heart. Rather, it is the expression of a general instability phenomenon. Experimental data suggest that the structural changes induced by the instability may cause secondary damage to the wall of small arteries and arterioles in the form of endothelial hyperpermeability followed by local fibrinoid necrosis of the vascular wall.