not time-reversible shape changes, we show that net locomotion can be obtained. Swimming arises either by continuously modulating fore-aft asymmetric vesicle shapes, or by crossing a continuous shape-transition region and alternating between fore-aft asymmetric and fore-aft symmetric shapes. In addition, the calculated hydrodynamic efficiencies are shown to be similar to that of other common low Reynolds number propulsive mechanisms. Our paper is organized as follows. We begin with a general discussion of the practical realiza- tion of controlled shape-changing vesicles, in particular the relevant time scales, and the possible actuation mechanisms. Two classical curvature-mediated vesicle models (spontaneous curvature and bilayer coupling) are presented, and the formulations used for the shape calculation and the numerical fluid-interaction model are introduced. We then discuss examples of vesicle shape cycles that yield a swimming motion, examine the fluid flow that develops around the vesicles during their deformation cycles, and compute the corresponding swimming speeds and hydrodynamic efficiencies. II. A ROADMAP TO VESICLE LOCOMOTION A vesicle immersed in a viscous fluid experiences a highly coupled array of forces, such as those generated by membrane tension, internal pressure, membrane (bending) elasticity, and the surrounding viscous fluid dynamics. In the microscopic, viscous environments relevant to our consideration, the Reynolds number, Re, is very small: Re = ρ U$_{c}$L$_{c}$/µ ≪ 1, where ρ is the fluid density, µ is the fluid shear viscosity, and U$_{c}$ and L$_{c}$ are characteristic velocity and length scales of the vesicle. The fluid behavior at low Reynolds number is highly dependent upon the immersed boundary geometry, and the resultant forces include not only local, but also non-local responses to its motion. A general study of vesicle dynamics should take non-equilibrium shapes into account, as even simple liposomes that can be created in situ can interact relatively quickly with the environment. It is possible to design experiments where carefully constructed initial conditions and lipid species lead to equilibrated vesicle shapes that are non-trivial, but in order to apply morphological changes and induce locomotion, a reversible parameter-changing mechanism is desirable. For our first approach to vesicle swimming, we consider in this paper a “stiff membrane” regime. The characteristic time of membrane relaxation in a viscous fluid is given by t$_{rel}$ = µr 3 $_{0}$/κ , where κ is the elastic bending modulus of the membrane, and r$_{0}$ is a characteristic radius of curvature. If we choose the maximum radius of the vesicle for the characteristic length scale L$_{c}$ , then r$_{0}$ ≲ L$_{c}$ . 3