Unsteady, viscous, low-speed flows over a NACA 0012 airfoil oscillated in plunge and/or pitch at various reduced frequency, amplitude, and phase shift are computed. Vortical wake formations, boundary-layer flows at the leading edge, the formation of leading-edge vortices and their downstream convection are presented in terms of unsteady particle traces. Flow separation characteristics and thrust-producing wake profiles are identified. Computed results compare well with water tunnel flow visualization and force data and other computational data. The maximum propulsive efficiency is obtained for cases where the flow remains mostly attached over the airfoil oscillated in a combined pitch and plunge.