Characteristic function of M. Liv\v{s}ic and some developments

The area related to M. Liv\v{s}ic's characteristic matrix functions is too vast to be discussed in one paper and we selected for this article the problems which are close to our scientific interests. We discuss M.Liv\v{s}ic's results connected with characteristic matrix functions and various important developments including factorization of the transfer operator function, inverse problems, triangular models, reduction of the operators to the simplest form as well as applications to Wiener--Masani prediction theory, to random matrices, and to Riemann--Hilbert problems.