Unsteady, viscous flows over flapping airfoils in a biplane configuration are computed on moving overset grids. The overset grid solutions are obtained in parallel in a distributed memory environment. Unsteady flowfields are described by particle traces: Time-averaged thrust values are obtained from the integration of the unsteady drag coefficient. It is shown that airfoils in a biplane configuration and oscillating in a combined pitch and plunge motion with a proper phase shift between them produce 20-40% more thrust than a single flapping airfoil. Turbulence in the flow further augments the thrust generation. For a maximum thrust at a given flapping frequency, an optimization of the flapping motion parameters is needed.