There has been a dramatic increase in the placement and use of arteriovenous fistulas (AVF) in US patients with chronic kidney disease over the past few years, in accordance with strong recommendations by Fistula First Initiative and KDOQI guidelines. However, AVF nonmaturation remains a substantial obstacle to achieving functional AVFs in a subset of patients, despite the widespread use of preoperative vascular mapping to assist surgeons in planning access surgery, and the growing use of interventions to salvage nonmaturing AVFs. In the right patient, aggressive efforts result in a functioning AVF, which provides adequate dialysis with relatively few interventions required to maintain its long-term patency for dialysis. In the wrong patient, aggressive efforts to achieve a mature AVF may result in numerous failed surgical and percutaneous procedures and prolonged catheter dependence, with all its associated complications. Thus, strict recommendations to place an AVF in all dialysis patients might not benefit every patient, and may actually harm some patients. There are no randomized clinical trials to address which patients are more suitable for placement of an arteriovenous graft (AVG), rather than an AVF. However, there is a wealth of observational studies, which taken cumulatively, may assist clinicians in identifying those patients who should receive an AVG. In this article, we review the relevant published literature regarding this topic and provide suggestions for stratifying patients who should receive each type of vascular access.