This is a survey of fractional calculus based on the fractional derivatives of the form
They coincide with the usual Riemann-Liouville derivatives up to finite-dimensional terms. Various known properties of such derivatives are presented together with their further developments and a number of applications.
Historical remark: Derivatives of form (1) and more general ones were first introduced and studied by M. Dzherbashyan and A. Nersesyan [Dokl. Akad. Nauk SSSR 121, 210–213 (1958; Zbl 0095.08504); Izv. Akad. Nauk Arm. SSR, Ser. Fiz.-Mat. Nauk 11, No.5, 85–106 (1958; Zbl 0086.05701)].