  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.
