Arterial grafts first appeared in the early 1950s, since when they have attracted much attention and undergone significant changes in structure and treatment. Arterial prostheses are now routinely used to correct arterial flow disorders. The success of large-diameter synthetic arteries has not been experienced in with small-caliber vessels. A major reason is the lack of transverse compliance, leading to mismatch in elastic stretch at the anastomoses and to failure. The chapter discusses the factors affecting compliance and methods of measuring and modeling it. Case studies involving prototyping of small diameter grafts and evaluation of their mechanical properties are described. Design principles for small-diameter grafts with desired specifications and performance characteristics, including compliance, are summarized.