An optimum aerodynamic design method has been developed for the wind-lens turbine. The wind-lens turbine has a brimmed diffuser around a turbine rotor, which is referred to as wind-lens. The wind-lens can achieve the wind concentration on the turbine rotor, resulting in the significant enhancement of the turbine output. The present design method is based on a quasi-three-dimensional aerodynamic design method and a genetic algorithm. The quasi-three-dimensional design consists of two parts: a meridional viscous flow analysis and two-dimensional blade element designs. In the meridional viscous flow analysis, the axisymmetric Reynolds-averaged Navier-Stokes equations are numerically solved on a meridional plane to determine the wind flow rate through the wind-lens and the spanwise flow distribution at the rotor. The turbine rotor blade geometry is determined by the two-dimensional blade element theory based on the momentum theorem of the ducted turbine. The turbine rotor and wind-lens are simultaneously optimized by the present design method. Aerodynamic performance and flow fields in the optimum and conventional design cases have been investigated by wind tunnel tests and three-dimensional Reynolds-averaged Navier-Stokes simulations, in order to verify the effectiveness of the present design method. It is found that the optimum design case achieves the significant improvement in the output power coefficient, so that its numerical and experimental results of the output power coefficient exceed the Betz limit, which is the theoretical maximum output power coefficient for bare wind turbines. It is revealed that the aerodynamic matching between the turbine rotor and the wind-lens is essential to the performance enhancement of the wind-lens turbine.