  This paper was published in the special issue of the Journal of Inequalities
and Special Functions dedicated to Professor Ivan Dimovski's contributions to
different fields of mathematics: transmutation theory, special functions,
integral transforms, function theory etc.
  This short survey paper contains brief historical information, main known
facts and original author's results on the theory of the Buschman-Erd\'{e}lyi
transmutations and some of their applications. The operators of
Buschman-Erd\'{e}lyi type were first studied by E.T.~Copson, R.G.~Buschman and
A.~Erd\'{e}lyi as integral operators. In 1990's the author was first to prove
the transmutational nature of these operators and published papers with
detailed study of their properties. This class include as special cases such
famous objects as the Sonine-Poisson-Delsarte transmutations and the fractional
Riemann-Lioville integrals. In this paper, the Buschman-Erdelyi transmutations
are logically classified as operators of the first kind with special case of
zero order smoothness operators, second kind and third kind with special case
of unitary Sonine-Katrakhov and Poisson-Katrakhov transmutations. We study such
properties as transmutational conditions, factorizations, norm estimates,
connections with classical integral transforms. Applications are considered to
singular partial differential equations, embedding theorems with sharp
constants in Kipriyanov spaces, Euler-Poisson-Darboux equation including Copson
lemma, generalized translations, Dunkl operators, Radon transform, generalized
harmonics theory, Hardy operators, V.\,Katrakhov's results on
pseudodifferential operators and boundary-value problems of new kind for
equations with solutions of arbitrary growth at the isolated singularity for
elliptic partial differential equations.
