In response to intracellular stress events ranging from starvation to pathogen invasion, the cell activates one or more forms of macroautophagy. The key event in these related pathways is the de novo formation of a new organelle called the autophagosome, which surrounds and sequesters either random portions of the cytoplasm or selectively targets individual intracellular challenges. Thus the autophagosome is a flexible membrane platform with dimensions that ultimately depend upon the target cargo. The intermediate membrane, termed the phagophore or isolation membrane, is a cup-like structure with a clear concave face and a highly curved rim. The phagophore is largely devoid of integral membrane proteins, thus its shape and size are governed by peripherally-associated membrane proteins and possibly by the lipid composition of the membrane itself. Growth along the phagophore rim marks the progress of both organelle expansion and ultimately of organelle closure around a particular cargo. These two properties, a reliance on peripheral membrane proteins and a structurally-distinct membrane architecture, suggest that the ability to target or manipulate membrane curvature might be an essential activity of proteins functioning in this pathway. In this review, we discuss the extent to which membranes are naturally curved at each of the cellular sites believed to engage in autophagosome formation, review basic mechanisms used to sense this curvature and then summarize the existing literature concerning which autophagy proteins are capable of curvature recognition.