The monograph sets out the foundations of the theory of interpolation of functions of matrix variables: the main problems are formulated, interpolation formulas are built for functions given on sets of square, rectangular matrices, including on sets of functional and random matrices. The problem of interpolation of functions of many matrix variables is considered, some variants of splines are proposed. Classes of matrix polynomials invariant under some of the constructed approximate interpolation-type formulas are indicated. A large number of examples are given for the construction of interpolation formulas and some other content.