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The word robust has been used in many contexts in signal processing. Our treatment concerns statistical robustness, which deals with deviations from the distributional assumptions. Many problems encountered in engineering practice rely on the Gaussian distribution of the data, which in many situations is well justified. This enables a simple derivation of optimal estimators. Nominal optimality, however, is useless if the estimator was derived under distributional assumptions on the noise and the signal that do not hold in practice. Even slight deviations from the assumed distribution may cause the estimator's performance to drastically degrade or to completely break down. The signal processing practitioner should, therefore, ask whether the performance of the derived estimator is acceptable in situations where the distributional assumptions do not hold. Isn't it robustness that is of a major concern for engineering practice? Many areas of engineering today show that the distribution of the measurements is far from Gaussian as it contains outliers, which cause the distribution to be heavy tailed. Under such scenarios, we address single and multichannel estimation problems as well as linear univariate regression for independently and identically distributed (i.i.d.) data. A rather extensive treatment of the important and challenging case of dependent data for the signal processing practitioner is also included. For these problems, a comparative analysis of the most important robust methods is carried out by evaluating their performance theoretically, using simulations as well as real-world data.