In this survey we briefly illustrate some of our contributions to the generalizations of the fractional calculus (analysis) as a theory of the operators for integration and differentiation of arbitrary (fractional) order, of the classical special functions and of the integral transforms of Laplace type. It is shown that these three topics of analysis are closely related and mutually induce their origins and developments. Due to the short space, we confine here only to survey the ideas of our recent contributions related to the title. Statements of the numerous results, their proofs, examples and applications can be found in Refs, like: [1]–[2], [5]–[7], [11]–[19].