  Recent results on the construction and applications of the transmutation
(transformation) operators are discussed. Three new representations for
solutions of the one-dimensional Schr\"odinger equation are considered. Due to
the fact that they are obtained with the aid of the transmutation operator all
the representations possess an important for practice feature. The accuracy of
the approximate solution is independent of the real part of the spectral
parameter. This makes the representations especially useful in problems
requiring computation of large sets of eigendata with a nondeteriorating
accuracy.
  Applications of the exact representations for the transmutation operators to
partial differential equations are discussed as well. In particular, it is
shown how the methods based on complete families of solutions can be extended
onto equations with variable coefficients.
