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This monograph opens with an introduction to the theory of probability metrics. The introduction is followed by some results based on the method of characteristic functions, the method of compositions and, finally, the method of metric distances. The idea of this method consists of the use of metrics that possess some special properties. The author considers the case of real-valued random variables, the case of random variables with values in finite-dimensional Euclidean spaces, and the case where the random variables take values in infinite-dimensional real separable Hilbert spaces.