Numerous animals including fish, birds, bats and insects use appendages (fins and wings) with varying levels of flexibility, to provide propulsion and/or weight support as they move through either air or water. Scientists have explored many different aspects of biological swimming and flying, such as flapping and flexibility effects, in an attempt to understand the underlying physical mechanisms and to apply them to engineering applications. Compliant membranes, used by bats, flying squirrels, lemurs and other animals define a specific subset of this class of flexible aerodynamic structures and is of particular interest due to their negligible thickness and large shape deformations in both steady and unsteady conditions. This work deals with the aerodynamic and hydrodynamic characteristics of a flapping, compliant membrane wing. A theoretical framework is developed to characterize the effects of wing compliance, inertia and flapping kinematics on the membrane’s performance. As the flapping frequency is increased, membranes go through a transition from thrust to drag around the resonant frequency, and this transition is earlier for more compliant membranes. The limitations of the linear, small-amplitude model are evaluated experimentally, by testing different compliant membranes performing a heaving motion in a water flume. Membrane deformations and force data show a resonant peak close to the natural frequency of the membranes. Thrust is evaluated through direct measurement as well as wake analysis, and the limitations of both methods are discussed. The analytic model and the experimental data provide guidelines to finding membrane and motion parameters to optimize the propulsive performance.