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Ever since its conception, the mean-squared error (MSE) criterion has been the workhorse of optimal linear adaptive filtering. However, it is a well-known fact that the MSE criterion is no longer optimal in situations where the data are corrupted by noise. Noise, being omnipresent in most of the engineering applications, can result in severe errors in the solutions produced by the MSE criterion. In this dissertation, we propose novel error criteria and the associated learning algorithms followed by a detailed mathematical analysis of these algorithms. Specifically, these criteria are designed to solve the problem of optimal filtering with noisy data. Firstly, we discuss a new criterion called augmented error criterion (AEC) that can provide unbiased parameter estimates even in the presence of additive white noise. Then, we derive novel, online sample-by-sample learning algorithms with varying degrees of complexity and performance that are tailored for real-world applications. Rigorous mathematical analysis of the new algorithms is presented. In the second half of this dissertation, we extend the AEC to handle correlated noise in the data. The modifications introduced will enable us to obtain optimal, unbiased parameter estimates of a linear system when the data are corrupted by correlated noise. Further, we achieve this without explicitly assuming any prior information about the noise statistics. The analytical solution is derived and an iterative stochastic algorithm is presented to estimate this optimal solution. The proposed criteria and the learning algorithms can be applied in many engineering problems. System identification and controller design problems are obvious areas where the proposed criteria can be efficiently used. Other applications include model-order estimation in the presence of noise and design of multiple local linear filters to characterize complicated nonlinear systems.