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In this article we consider theorems on expansions of iterated Ito andStratonovich stochastic integrals. These theorems open up a new direction forstudy of iterated Ito and Stratonovich stochastic integrals. The expansionsbased on multiple and iterated Fourier-Legendre series as well as on multipleand iterated trigonomectic Fourier series converging in the mean and pointwiseare presented in the article. Some of these theorems are connected with theiterated stochastic integrals of multiplicities $1-6$. Also we consider twotheorems on expansions of iterated Ito stochastic integrals of arbitrarymultiplicity $k$ $(k\in\mathbb{N})$ based on generalized multiple Fourierseries converging in the sense of norm in Hilbert space $L_2([t, T]^k)$ as wellas two theorems on expansions of iterated Stratonovich stochastic integrals ofarbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalized iteratedFourier series converging pointwise. On the base of the presented theorems weformulate three Hypotheses on expansions of iterated Stratonovich stochasticintegrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalizedmultiple Fourier series converging in the sense of norm in Hilbert space$L_2([t, T]^k).$ The proof of Hypothesis $2$ is given under the condition ofconvergence of trace series. The mentioned iterated Stratonovich stochasticintegrals are part of the Taylor-Stratonovich expansion. Moreover, most of theconsidered expansions of iterated Stratonovich stochastic integrals containonly one operation of the limit transition and substantially simpler than theiranalogues for iterated Ito stochastic integrals. The results of the article canbe applied to numerical integration of Ito stochastic differential equations.Also, some results of the article were reformulated in the form of theorems ofthe Wong-Zakai type for iterated Stratonovich stochastic integrals.

The Hypotheses on Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity and Their Partial Proof