Two optimal beamforming approaches for sensor arrays with arbitrary shapes and element directivities are proposed. The first one minimizes the sidelobe level while keeping the distortionless response in the direction of the desired signal and maintaining the mainlobe width. The second approach maximizes the array gain while keeping the mainlobe direction and controling the sidelobe level which is strictly guaranteed to be below a prescribed value. Array weight norm constraint is used to improve the robustness of the two optimal beamforming approaches against random errors. The first approach provides the optimal tradeoff among the sidelobe level, the beamwidth and robustness; and the second approach provides the optimal tradeoff among the array gain, the sidelobe level and robustness. Both optimal beamforming problems are formulated as the second-order cone programming which can be easily solved using well-developed interior-point methods. Results of computer simulations and lake-experiment for a circula Two optimal beamforming approaches for sensor arrays with arbitrary shapes and element directivities are proposed. The first one minimizes the sidelobe level while keeping the distortionless response in the direction of the desired signal and maintain the mainlobe width. The second approach maximizes the array gain while keeping the mainlobe direction and controling the sidelobe level which is strictly guaranteed to be below a prescribed value. Array weight norm constraint is used to improve the robustness of the two optimal beamforming approaches against random errors. The first principle provides the optimal tradeoff among the sidelobe levels , the beamwidth and robustness; and the second approach provide the optimal tradeoff among the array gain, the sidelobe level and robustness. Both optimal beamforming problems are formulated as the second-order cone programming which can be easily solved using well-developed interior-point methods. Results of computer simulations and lake-exper iment for a circula