A brief history and exposition of the fundamental theory of fractional calculus.- The use in mathematical physics of Erdélyi-Kober operators and of some of their generalizations.- The weyl fractional calculus.- H-R transform in two dimensions and some of its applications to partial differential equations.- Inequalities via fractional integration.- An access to fractional differentiation via fractional difference quotients.- A family of integral representations for the solution of the diffusion equation.- Fractional integrals of generalized functions.- The fractional derivative and entire functions.- Formulas of the dirichlet-mehler type.- A child's garden of special functions.- An algebraic definition of fractional differentiation.- Generalized poisson integrals and regularity of functions.- Fractional spaces of temperate distribution.- Applications of fractional calculus to spherical (radial) probability models and generalizations.- A problem of hyperstereology.- A hypergeometric integral equation.- Application of fractional differentiation to the modeling of hodograph linearities.- Fractional calculus in the operator field of generalized functions.- A functional relation.- On moments of probability distribution functions.- Fractional integration of fundamental solutions.- Fundamental properties of fractional derivatives via pochhammer integrals.- On the recent trends in the development, theory and applications of fractional calculus.- Open questions for research.