  This is an extended version of originally published survey in the book:
"Advances in Modern Analysis and Mathematical Modeling". Editors:
Yu.F.Korobeinik, A.G.Kusraev. Vladikavkaz: Vladikavkaz Scientific Center, of
the Russian Academy of Sciences and Republic of North Ossetia--Alania, 2008. P.
226-293. (In Russian).
  In this survey we consider main transmutation theory topics with many
applications, including author's own results. The topics covered are:
transmutations for Sturm--Liouville operators, Vekua-Erdelyi-Lowndes
transmutations, transmutations for general differential operators with variable
coefficients, Sonine and Poisson transmutations, transmutations and fractional
integrals, Buschman--Erdelyi transmutations, in the search for Volterra unitary
transmutations, transmutations for singular differential operators with
variable coefficients, composition method for transmutations, some applications
and open problems.
