  The book is devoted to the strong approximation of iterated stochastic
integrals (ISIs) in the context of numerical integration of Ito SDEs and
non-commutative semilinear SPDEs with nonlinear multiplicative trace class
noise. The monograph opens up a new direction in researching of ISIs. For the
first time we successfully use the generalized multiple Fourier series
converging in the sense of norm in $L_2([t, T]^k)$ for the expansion and strong
approximation of Ito ISIs of multiplicity $k,$ $k\in{\bf N}$ (Chapter 1). This
result has been adapted for Stratonovich ISIs of multiplicities 1 to 6 (the
case of continuously differentiable weight functions and a CONS of Legendre
polynomials or trigonometric functions in $L_2([t, T])$) and for Stratonovich
ISIs of multiplicities 1 to 5 (the case of continuous weight functions and an
arbitrary CONS in $L_2([t, T])$) (Chapter 2), as well as for some other types
of iterated stochastic integrals (Chapter 1). Recently (in 2024), the mentioned
adaptation has also been carried out for Stratonovich ISIs of multiplicity $k$
$(k\in{\bf N})$ for the case of continuous weight functions and an arbitrary
CONS in $L_2([t, T])$ (Theorem 2.61). We derived the exact and approximate
expressions for the mean-square error of approximation of Ito ISIs of
multiplicity $k$, $k\in{\bf N}$ (Chapter 1). We provided a significant
practical material (Chapter 5) devoted to the expansions of specific Ito and
Stratonovich ISIs of multiplicities 1 to 6 from the Taylor-Ito and
Taylor-Stratonovich expansions (Chapter 4) using the CONS of Legendre
polynomials and the CONS of trigonometric functions. The methods formulated in
this book have been compared with some existing methods (Chapter 6). The
results of Chapter 1 were applied (Chapter 7) to the approximation of ISIs of
multiplicity $k,$ $k\in{\bf N}$ with respect to the $Q$-Wiener process.
