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We consider some versions and generalizations of an approach to the expansionof iterated Ito stochastic integrals of arbitrary multiplicity $k$$(k\in\mathbb{N})$ based on generalized multiple Fourier series. Expansions ofiterated stochastic integrals with respect to martingale Poisson measures andwith respect to martingales were obtained. For the iterated stochasticintegrals with respect to martingales we have proved theorem, which is ageneralization of the expansion for iterated Ito stochastic integrals ofarbitrary multiplicity based on generalized multiple Fourier series. Also weconsider a modification of the mentioned expansion of iterated Ito stochasticintegrals for the case of complete orthonormal with weight $r(t_1)\ldotsr(t_k)\ge 0$ systems of functions in the space $L_2([t, T]^k)$. Mean-squareconvergence of the considered expansions is proved. An example of the expansionof iterated (double) stochastic integrals with respect to martingales using thesystem of Bessel functions is considered.

Expansion of Iterated Stochastic Integrals with Respect to Martingale  Poisson Measures and with Respect to Martingales Based on Generalized  Multiple Fourier Series