  This paper is partly a historical survey of various approaches and methods in
the fractional calculus, partly a description of the Kipriyanov extraordinary
theory in comparison with the classical one. The significance and outstanding
methods in constructing the independent Kipriyanov fractional calculus theory
are convexly stressed, also we represent modern results involving the
Kipriyanov operator and corresponding generalization under the semigroup theory
point of view.
