Abstract: Two optimal beamforming approaches for sensor arrays of arbitrary geometry and element directivity are developed. The first one minimizes the sidelobe level while keeping the distortionless response in the direction of desired signal and maintaining the mainlobe width. This approach can obtain lower sidelobe level compared to other adaptive-based low-sidelobe beamforming algorithms while keeping the same beamwidth. The second approach maximizes the array gain while keeping mainlobe direction and controling the maximum sidelobe level which is strictly guaranteed to be lower than a prescribed value. The approach can obtain higher array gain compared to the adaptive-based low-sidelobe beamforming algorithms while keeping the same sidelobe level. Array weight norm constraint is used to increase the robustness of the two optimal beamforming against random errors. Both of the optimal beamforming problems are converted to a convex form as the so-called second-order cone programming and easily solved using well-developed interior-point methods.Resultsof computer simulation and lake-experiment for a circular array show superior performance of the proposed approaches developed in this paper in comparison to the existing adaptive-based algorithms.