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This is a survey paper in two parts. In the first part we list main variantsof one-dimensional fractional integrodifferential operators. Also somehistorical and priority remarks are given. As a special question we considerthe impact of Soviet and Russian researches to fractional calculus and itsapplications to viscoelasticity theory. We also stress an impact of theVoronezh school of mechanics and viscoelasticity theory, including works ofShermergor, Meschkov, Rossikhin, Shitikova and others. In the second part ofthe paper we consider two important special classes of generalized fractionaloperators, they were thoroughly studied by the authors. We considerBuschman-Erdelyi operators, this is an important class containing as specialcases Riemann-Liouville, Erd\'elyi-Kober, Mehler-Fock operators and someothers. They also include classical transmutations for the Bessel differentialoperator, namely Sonine and Poisson ones. After that we represent results onanother important generalization of Riemann-Liouville operators, namelyfractional powers of the Bessel operators. The paper is dedicated to AdamMaremovich Nakhushev -- brilliant man and mathematician.

On two classes of generalized fractional operators (with short historical survey of fractional calculus)