  We start with the Birman--Solomyak approach to define double operator
integrals and consider applications in estimating operator differences
$f(A)-f(B)$ for self-adjoint operators $A$ and $B$. We present the
Birman--Solomyak approach to the Lifshits--Krein trace formula that is based on
double operator integrals. We study the class of operator Lipschitz functions,
operator differentiable functions, operator H\"older functions, obtain
Schatten--von Neumann estimates for operator differences. Finally, we consider
in Chapter 1 estimates of functions of normal operators and functions of
$d$-tuples of commuting self-adjoint operators. In Chapter 2 we define multiple
operator integrals with integrands in the integral projective tensor product of
$L^\infty$ spaces. We consider applications of such multiple operator integrals
to the problem of the existence of higher operator derivatives and to the
problem of estimating higher operator differences. We also consider connections
with trace formulae for functions of operators under perturbations of class
$\boldsymbol{S}_m$, $m\ge2$. In the last chapter we define Haagerup-like tensor
products of the first kind and of the second kind and we use them to study
functions of noncommuting self-adjoint operators under perturbation. We show
that for functions $f$ in the Besov class $B_{\infty,1}^1({\Bbb R}^2)$ and for
$p\in[1,2]$ we have a Lipschitz type estimate in the Schatten--von Neumann norm
$\boldsymbol{S}_p$ for functions of pairs of noncommuting self-adjoint
operators, but there is no such a Lipschitz type estimate in the norm of
$\boldsymbol{S}_p$ with $p>2$ as well as in the operator norm. We also use
triple operator integrals to estimate the trace norms of commutators of
functions of almost commuting self-adjoint operators and extend the
Helton--Howe trace formula for arbitrary functions in the Besov space
$B_{\infty,1}^1({\Bbb R}^2)$.
