This book provides a comprehensive treatment of multilinear operator integral techniques. The exposition is structured to
be suitable for a course on methods and applications of multilinear operator integrals and also as a research aid. The ideas
and contributions to the field are surveyed and up-to-date results and methods are presented. Most practical constructions
of multiple operator integrals are included along with fundamental technical results and major applications to smoothness
properties of operator functions (Lipschitz and Hoelder continuity, differentiability), approximation of operator functions,
spectral shift functions, spectral flow in the setting of noncommutative geometry, quantum differentiability, and differentiability
of noncommutative L^p-norms. Main ideas are demonstrated in simpler cases, while more involved, technical proofs are outlined
and supplemented with references. Selected open problems in the field are also presented.