This paper presents an overview of the applications of fractional calculus in continuum and statistical mechanics, focusing on linear viscoelasticity, the Basset problem, Brownian motion, and the fractional diffusion-wave equation. The author, Francesco Mainardi, discusses how fractional calculus provides a powerful tool for modeling complex systems with memory and hereditary properties. The paper is based on lectures given at a CISM course on scaling laws and fractality in continuum mechanics, and it includes a revised version of a chapter from a book on fractals and fractional calculus in continuum mechanics. The paper begins with an introduction to linear viscoelasticity, explaining the concepts of creep compliance and relaxation modulus, and how fractional calculus can be used to generalize classical models. It then discusses the mechanical models used in viscoelasticity, including the Hooke, Newton, Voigt, and Maxwell models, and how fractional calculus can be applied to these models to derive generalized forms. The paper then explores the Basset problem, which involves the motion of a particle in a viscous fluid. The author introduces the generalized Basset force, which is expressed in terms of a fractional derivative of any order between 0 and 1. This generalization allows for a more accurate description of the long-time behavior of the solution, changing the algebraic decay from $ t^{-1/2} $ to $ t^{-\alpha} $. The paper also discusses Brownian motion and its connection to fractional calculus. It explains how fractional calculus can be used to model the long tails in the velocity correlation and displacement variance, which are characteristic of anomalous diffusion. The author introduces the fractional diffusion-wave equation, which is obtained from the classical diffusion equation by replacing the first-order time derivative with a fractional derivative of order $ \beta $, where $ 0 < \beta < 2 $. The fundamental solutions of this equation are expressed in terms of two interrelated auxiliary functions, which are of Wright type. The paper concludes with a discussion of the applications of fractional calculus in modeling various phenomena, including the use of the Mittag-Leffler function in the analysis of fractional relaxation processes. The author also references several other works that have explored the applications of fractional calculus in physics and engineering.This paper presents an overview of the applications of fractional calculus in continuum and statistical mechanics, focusing on linear viscoelasticity, the Basset problem, Brownian motion, and the fractional diffusion-wave equation. The author, Francesco Mainardi, discusses how fractional calculus provides a powerful tool for modeling complex systems with memory and hereditary properties. The paper is based on lectures given at a CISM course on scaling laws and fractality in continuum mechanics, and it includes a revised version of a chapter from a book on fractals and fractional calculus in continuum mechanics. The paper begins with an introduction to linear viscoelasticity, explaining the concepts of creep compliance and relaxation modulus, and how fractional calculus can be used to generalize classical models. It then discusses the mechanical models used in viscoelasticity, including the Hooke, Newton, Voigt, and Maxwell models, and how fractional calculus can be applied to these models to derive generalized forms. The paper then explores the Basset problem, which involves the motion of a particle in a viscous fluid. The author introduces the generalized Basset force, which is expressed in terms of a fractional derivative of any order between 0 and 1. This generalization allows for a more accurate description of the long-time behavior of the solution, changing the algebraic decay from $ t^{-1/2} $ to $ t^{-\alpha} $. The paper also discusses Brownian motion and its connection to fractional calculus. It explains how fractional calculus can be used to model the long tails in the velocity correlation and displacement variance, which are characteristic of anomalous diffusion. The author introduces the fractional diffusion-wave equation, which is obtained from the classical diffusion equation by replacing the first-order time derivative with a fractional derivative of order $ \beta $, where $ 0 < \beta < 2 $. The fundamental solutions of this equation are expressed in terms of two interrelated auxiliary functions, which are of Wright type. The paper concludes with a discussion of the applications of fractional calculus in modeling various phenomena, including the use of the Mittag-Leffler function in the analysis of fractional relaxation processes. The author also references several other works that have explored the applications of fractional calculus in physics and engineering.