Based on the traditional Minimum Mean Square Error (MMSE) criterion we commence our study by investigating three classical algorithms, namely the Recursive Least Square (RLS), the Direct Matrix Inversion (DMI) and the Recursive Sample Matrix Inversion (RSMI) algorithms in order to determine the optimal array weight values. We evaluate and compare their performance in terms of the attainable Signal-to-Interference Ratio (SIR), as a function of the received signal power and that of the reference sequence length used for Binary Phase Shift Keying (BPSK) transmissions over an Additive White Gaussian Noise (AWGN) channel. A three-element uniform linear array was then employed to observe the array's angular response at the defined angles, as the number of interfering signals is increased. The RLS algorithm was used for conducting these simulations over both AWGN and flat Rayleigh fading channels. For the adaptive algorithms discussed, we evaluate the associated complexity as a function of the number of antenna elements invoked. Our further investigations were motivated by the rationale that the ultimate performance measure is the achievable bit error ratio. Therefore, instead of the MMSE criterion we continue our investigations by invoking the novel approach of Minimum Bit Error Ratio (MBER) beamforming, which is based on directly minimising the system's Bit Error Ratio (BER). We employed a simplified conjugate gradient algorithm for determining the array weights of this MBER beamforming solution. Several adaptive versions of the MBER algorithm were presented, which were categorised into two classes, namely the family of block-data based and the set of sample-by-sample adaptive stochastic gradient based algorithms. In the block-data based adaptive algorithm category we studied the Block Adaptive Conjugate Gradient (BACG) algorithm, while in the stochastic gradient category we investigated both the Least Bit Error Rate (LBER) and the Approximate LBER (ALBER) algorithms. The ALBER algorithm consistently outperformed the LBER algorithm, despite having as low a complexity as the well-known LMS algorithm, provided that the related algorithmic parameters were appropriately chosen. To circumvent the drawbacks of the gradient based adaptive MBER algorithms, namely those of the BACG, LBER and ALBER algorithms we invoked Genetic Algorithms (GAs) in conjunction with the MBER beamforming scheme. The convergence behaviour of the GA was studied by evaluating the probability density function (PDF) of the BER at the beamformer's output. It was shown that GA is capable of circumventing many of the problems encountered by the MBER beamforming scheme.