The paper presents an introduction to fractional calculus, focusing on the Riemann-Liouville approach to fractional integration and differentiation. It discusses the Laplace transform technique for solving fractional integral and differential equations, emphasizing the Mittag-Leffler function. The key topics include the essentials of Riemann-Liouville fractional calculus, Abel-type integral equations, and relaxation and oscillation equations of fractional order. The paper also explores the definitions and properties of fractional derivatives, including the Caputo fractional derivative, and their Laplace transform representations. It addresses the semigroup property of fractional integrals, the behavior of fractional derivatives of power functions, and the differences between Riemann-Liouville and Caputo fractional derivatives. The text also covers other definitions and notations in fractional calculus, the law of exponents for fractional operators, and applications of fractional integral equations, including Abel equations of the first and second kind. The paper concludes with an analysis of fractional differential equations, focusing on simple fractional relaxation and oscillation equations, and their solutions using the Caputo fractional derivative. The work provides a comprehensive overview of fractional calculus, its mathematical foundations, and its applications in various fields.The paper presents an introduction to fractional calculus, focusing on the Riemann-Liouville approach to fractional integration and differentiation. It discusses the Laplace transform technique for solving fractional integral and differential equations, emphasizing the Mittag-Leffler function. The key topics include the essentials of Riemann-Liouville fractional calculus, Abel-type integral equations, and relaxation and oscillation equations of fractional order. The paper also explores the definitions and properties of fractional derivatives, including the Caputo fractional derivative, and their Laplace transform representations. It addresses the semigroup property of fractional integrals, the behavior of fractional derivatives of power functions, and the differences between Riemann-Liouville and Caputo fractional derivatives. The text also covers other definitions and notations in fractional calculus, the law of exponents for fractional operators, and applications of fractional integral equations, including Abel equations of the first and second kind. The paper concludes with an analysis of fractional differential equations, focusing on simple fractional relaxation and oscillation equations, and their solutions using the Caputo fractional derivative. The work provides a comprehensive overview of fractional calculus, its mathematical foundations, and its applications in various fields.