This book is the first monograph where the problem of strong (mean-square) approximation of multiple Ito and Stratonovich stochastic integrals is sistematically analyzed in the context of numerical integration of stochastic differential Ito equations. This monograph for the first time successfully use the tool 2 of multiple and iterative Fourier series, built in the space L2 and poitwise, for the strong approximation of multiple stochastic integrals. The aforesaid means were not used in this academic field before. 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 theorem on expansion of multiple Stratonovich stochastic integrals with any fixed multiplicity k, based on generalized Fourier series converging pointwise is verified. We obtained exact expressions for mean-square errors of approximation of multiple stochastic Ito integrals of 1 -- 4 multiplicity. We provided a significant practical material devoted to expansion and approximation of specific multiple Ito and Stratonovich stochastic integrals of 1 -- 5 multiplicity using the system of Legendre polynomials and the system of trigonometric functions. We compared the methods formulated in this book with existing methods of strong approximation of multiple stochastic integrals. This monograph open a new direction in researching of multiple Ito and Stratonovich stochastic integrals. This book will be interesting for specialists dealing with the theory of stochastic processes, applied and computational mathematics, senior students and postgraduates of technical institutes and universities, as well as for computer experts.