In this thesis, we present a Bayesian approach to the problem of beamforming without accurate knowledge about the direction-of-arrival. Under the Bayesian formulation, the proposed beamformer is constructed as a mixture of directional beamformers combined according to the data-driven posterior distribution. As the number of recevied data increases, the Bayesian beamformer asymptotically converges into a directional beamformer that points at the closest admissible direction to the true underlying direction, where closeness is defined in the Kullback Leibler divergence sense. The rate of convergence of the beamformer is controlled by the signal-to-noise ratio (SNR) of the array. Three efficient implementation algorithms for the Bayesian beamformer are developed by exploiting the structure of the steering vector of a uniform linear array (ULA). Each algorithm is analogous to a conventional filter design technique. The strengths, weaknesses, and design tradeoffs of the three algorithms are discussed. The performance of the proposed beamformer is compared to other existing robust beamformers via numerical simulations.