Biological membranes exhibit various function-related shapes, and the mechanism by which these shapes are created is largely unclear. Here, we classify possible curvature-generating mechanisms that are provided by lipids that constitute the membrane bilayer and by proteins that interact with, or are embedded in, the membrane. We describe membrane elastic properties in order to formulate the structural and energetic requirements of proteins and lipids that would enable them to work together to generate the membrane shapes seen during intracellular trafficking.