Numerical procedures are presented for the systematic computation of unsteady flows over moving airfoils or airfoil combinations, and these procedures are applied to the investigation of flapping-wing propulsion and power extraction. Flow solutions about single foils are computed using an unsteady, two-dimensional panel code coupled with a boundary layer algorithm and driven using an interactive graphical user interface. Flow solutions about airfoil combinations are computed using a companion, multi-element version of the panel code. Results for pitching-only and plunging-only motions compare favorably with theory and reasonably well with experimental results. Extensive computations are performed over the broad parameter space for combined pitching and plunging motions using the foil as both a propulsive device and as a wingmill or power-extraction device. Results modeling flight in ground effect are compared with other numerical and experimental results.