This paper, based on lectures by Rudolf Gorenflo and Francesco Mainardi, introduces the linear operators of fractional integration and fractional differentiation within the framework of Riemann-Liouville fractional calculus. The authors focus on the technique of Laplace transforms to handle these operators, making the material accessible to applied scientists. They derive analytical solutions for simple linear integral and differential equations of fractional order and highlight the role of the Mittag-Leffler function. The topics covered include the essentials of Riemann-Liouville fractional calculus, Abel-type integral equations, and relaxation and oscillation differential equations of fractional order. The paper also discusses the Caputo fractional derivative, which incorporates initial values of the function and its integer derivatives, and explores the differences between the standard Riemann-Liouville fractional derivative and the Caputo fractional derivative. Additionally, it covers other definitions and notations, the law of exponents, and applications of Abel integral equations in various fields such as spectroscopic measurements, planetary atmosphere studies, and inverse boundary value problems in partial differential equations.This paper, based on lectures by Rudolf Gorenflo and Francesco Mainardi, introduces the linear operators of fractional integration and fractional differentiation within the framework of Riemann-Liouville fractional calculus. The authors focus on the technique of Laplace transforms to handle these operators, making the material accessible to applied scientists. They derive analytical solutions for simple linear integral and differential equations of fractional order and highlight the role of the Mittag-Leffler function. The topics covered include the essentials of Riemann-Liouville fractional calculus, Abel-type integral equations, and relaxation and oscillation differential equations of fractional order. The paper also discusses the Caputo fractional derivative, which incorporates initial values of the function and its integer derivatives, and explores the differences between the standard Riemann-Liouville fractional derivative and the Caputo fractional derivative. Additionally, it covers other definitions and notations, the law of exponents, and applications of Abel integral equations in various fields such as spectroscopic measurements, planetary atmosphere studies, and inverse boundary value problems in partial differential equations.