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The minimax approach to the design of systems that are robust with respect to modeling uncertainties is studied using a game theoretic formulation in which the performance functional and the sets of modeling uncertainties and admissible design policies are arbitrary. The existence and characterization of minimax robust solutions that form saddle points are discussed through various methods that take into account several common features of the games encountered in applications. In particular, it is shown that if the performance functional and the uncertainty set are convex then a certain type of regularity condition on the functional is sufficient to ensure that the optimal strategy for a least favorable element of the uncertainty set is minimax robust. The efficacy of the methods proposed for a general game is tested in the problems of matched filtering, Wiener filtering, quadratic detection, and output energy filtering, in which uncertainties in their respective signal and noise models are assumed to exist. These problems are analyzed in a common Hilbert space framework and they serve to point out the advantages and limitations of the proposed techniques.