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We give an overview of recent developments in Sturm-Liouville theoryconcerning operators of transmutation (transformation) and spectral parameterpower series (SPPS). The possibility to write down the dispersion(characteristic) equations corresponding to a variety of spectral problemsrelated to Sturm-Liouville equations in an analytic form is an attractivefeature of the SPPS method. It is based on a computation of certain systems ofrecursive integrals. Considered as families of functions these systems arecomplete in the $L_{2}$-space and result to be the images of the nonnegativeinteger powers of the independent variable under the action of a correspondingtransmutation operator. This recently revealed property of the Delsartetransmutations opens the way to apply the transmutation operator even when itsintegral kernel is unknown and gives the possibility to obtain furtherinteresting properties concerning the Darboux transformed Schr\"{o}dingeroperators.  We introduce the systems of recursive integrals and the SPPS approach,explain some of its applications to spectral problems with numericalillustrations, give the definition and basic properties of transmutationoperators, introduce a parametrized family of transmutation operators, studytheir mapping properties and construct the transmutation operators for Darbouxtransformed Schr\"{o}dinger operators.

Transmutations and spectral parameter power series in eigenvalue problems