  In the present review we deal with the recently introduced method of spectral
parameter power series (SPPS) and show how its application leads to an explicit
form of the characteristic equation for different eigenvalue problems involving
Sturm-Liouville equations with variable coefficients. We consider
Sturm-Liouville problems on finite intervals; problems with periodic potentials
involving the construction of Hill's discriminant and Floquet-Bloch solutions;
quantum-mechanical spectral and transmission problems as well as the eigenvalue
problems for the Zakharov-Shabat system. In all these cases we obtain a
characteristic equation of the problem which in fact reduces to finding zeros
of an analytic function given by its Taylor series. We illustrate the
application of the method with several numerical examples which show that at
present the SPPS method is the easiest in the implementation, the most accurate
and efficient. We emphasize that the SPPS method is not a purely numerical
technique. It gives an analytical representation both for the solution and for
the characteristic equation of the problem. This representation can be
approximated by different numerical techniques and for practical purposes
constitutes a powerful numerical method but most important it offers additional
insight into the spectral and transmission problems.
