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The performance of adaptive beamforming methods is known to degrade severely in the presence of even small mismatches between the actual and presumed array responses to the desired signal. Such mismatches may frequently occur in practical situations because of violation of underlying assumptions on the environment, sources, or sensor array. This is especially true when the desired signal components are present in the beamformer "training" data snapshots because in this case, the adaptive array performance is very sensitive to array and model imperfections. The similar phenomenon of performance degradation can occur even when the array response to the desired signal is known exactly, but the training sample size is small. We propose a new powerful approach to robust adaptive beamforming in the presence of unknown arbitrary-type mismatches of the desired signal array response. Our approach is developed for the most general case of an arbitrary dimension of the desired signal subspace and is applicable to both the rank-one (point source) and higher rank (scattered source/fluctuating wavefront) desired signal models. The proposed robust adaptive beamformers are based on explicit modeling of uncertainties in the desired signal array response and data covariance matrix as well as worst-case performance optimization. Simple closed-form solutions to the considered robust adaptive beamforming problems are derived. Our new beamformers have a computational complexity comparable with that of the traditional adaptive beamforming algorithms, while, at the same time, offer a significantly improved robustness and faster convergence rates.