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The book is devoted to the problem of mean-square approximation of multiple Ito and Stratonovich stochastic integrals in the context of numerical integration of stochastic differential Ito equations. The monograph successfully use the tool of multiple and iterative Fourier-Legendre and trigonometric Fourier series, built in the space L2 and poitwise, for the mean-square approximation of multiple stochastic integrals. We obtained a general result connected with expansion of multiple stochastic Ito integrals with any fixed multiplicity k, based on generalized multiple Fourier series converging in the space L2([t, T]x...x[t, T]) (x...x -- k-1 times). This result is adapted for multiple Stratonovich stochastic integrals of 1 -- 4 multiplicity for Legendre polynomial system and system of trigonometric functions, as well as for other types of multiple stochastic integrals. The monograph open a new direction in researching of multiple Ito and Stratonovich stochastic integrals.