Many cellular membrane-bound structures exhibit distinct curvature that is driven by the physical properties of their lipid and protein constituents. Here we review how cells manipulate and control this curvature in the context of dynamic events such as vesicle-mediated membrane traffic. Lipids and cargo proteins each contribute energy barriers that must be overcome during vesicle formation. In contrast, protein coats and their associated accessory proteins drive membrane bending using a variety of interdependent physical mechanisms. We survey the energy costs and drivers involved in membrane curvature, and draw a contrast between the stochastic contributions of molecular crowding and the deterministic assembly of protein coats. These basic principles also apply to other cellular examples of membrane bending events, including important disease-related problems such as viral egress.