  In this article, we collected more than thirty theorems on expansions of
iterated Ito and Stratonovich stochastic integrals, which have been formulated
and proved by the author in the period from 1997 to 2024. These theorems open
up a new direction for study of iterated Ito and Stratonovich stochastic
integrals. We consider two theorems on expansion of iterated Ito stochastic
integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalized
multiple Fourier series converging in the sense of norm in Hilbert space
$L_2([t, T]^k)$. We adapt these theorems on expansion of iterated Ito
stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ for
iterated Stratonovich stochastic integrals of multiplicities 1 to 6 (the case
of continuously differentiable weight functions and a complete orthonormal
system of Legendre polynomials or trigonometric functions in $L_2([t, T])$) and
for iterated Stratonovich stochastic integrals of multiplicities 1 to 5 (the
case of continuous weight functions and an arbitrary complete orthonormal
system of functions in $L_2([t, T])$). On the base of the presented theorems we
formulate several hypotheses on expansions of iterated Stratonovich stochastic
integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$. Recently,
Hypotheses 7 and 8 have been proved for the case of an arbitrary complete
orthonormal system of functions in $L_2([t, T]),$ $p_1=\ldots=p_k=p,$
$k\in\mathbb{N}$ (Theorems 56, 57) but under an additional assumption. The
considered expansions converge in the mean-square sense and contain only one
operation of the limit transition in contrast to its existing analogues. The
mentioned iterated Stratonovich stochastic integrals are part of the
Taylor-Stratonovich expansion, i.e. the results of the article can be applied
to numerical integration of Ito SDEs.
