The article is devoted to the mean-square approximation of iterated Ito and Stratonovich stochastic integrals in the context of the numerical integration of Ito stochastic differential equations. The expansion of iterated Ito stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ and expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 6 have been obtained. Considerable attention is paid to expansions based on multiple Fourier-Legendre series. The exact and approximate expressions for the mean-square error of approximation of iterated Ito stochastic integrals are derived. The results of the article will be useful for numerical integration of Ito stochastic differential equations with non-commutative noise.