The theory of intra-surface viscous flow on lipid bilayers is developed by combining the equations for flow on a curved surface with those that describe the elastic resistance of the bilayer to flexure. The model is derived directly from balance laws and augments an alternative formulation based on a variational principle. Conditions holding along an edge of the membrane are emphasized and the coupling between flow and membrane shape is simulated numerically. Particular applications that will be discussed in this talk include lipid flow on minimal surfaces, osmotic pressure drive membrane deformation, and flow due to protein adsorption.