The paper by Francesco Mainardi reviews the application of fractional calculus to fundamental problems in continuum and statistical mechanics. The main topics covered include: 1. **Linear Viscoelasticity and Fractional Calculus**: The paper discusses the mathematical modeling of viscoelastic bodies, introducing fractional calculus to generalize classical spring-dashpot models. It covers the fundamentals of linear viscoelasticity, mechanical models, and the introduction of fractional viscoelastic models, including the fractional S.L.S. (Standard Linear Solid) model. 2. **The Basset Problem via Fractional Calculus**: The dynamics of a sphere in a viscous fluid is analyzed, focusing on the Basset force, which is related to the history of the relative acceleration of the sphere. The paper introduces a generalized Basset force expressed as a fractional derivative of the particle velocity, and provides solutions for the particle velocity in terms of Mittag-Leffler functions. The effect of the generalized Basset term on the long-time behavior of the solution is discussed. 3. **Brownian Motion and Fractional Calculus**: The classical Langevin equation is revisited, and a hydrodynamic model that includes the Basset force is introduced. This model leads to a fractional Langevin equation, which accounts for the inertial effect and the retarding effect due to the Basset force. The paper explores the connection between fractional Brownian motion and anomalous diffusion. The paper also includes an appendix on the Wright function and references to recent developments in the field. The revised version includes corrections, improvements, and updated references, reflecting the author's ongoing research in the area of fractional calculus and its applications.The paper by Francesco Mainardi reviews the application of fractional calculus to fundamental problems in continuum and statistical mechanics. The main topics covered include: 1. **Linear Viscoelasticity and Fractional Calculus**: The paper discusses the mathematical modeling of viscoelastic bodies, introducing fractional calculus to generalize classical spring-dashpot models. It covers the fundamentals of linear viscoelasticity, mechanical models, and the introduction of fractional viscoelastic models, including the fractional S.L.S. (Standard Linear Solid) model. 2. **The Basset Problem via Fractional Calculus**: The dynamics of a sphere in a viscous fluid is analyzed, focusing on the Basset force, which is related to the history of the relative acceleration of the sphere. The paper introduces a generalized Basset force expressed as a fractional derivative of the particle velocity, and provides solutions for the particle velocity in terms of Mittag-Leffler functions. The effect of the generalized Basset term on the long-time behavior of the solution is discussed. 3. **Brownian Motion and Fractional Calculus**: The classical Langevin equation is revisited, and a hydrodynamic model that includes the Basset force is introduced. This model leads to a fractional Langevin equation, which accounts for the inertial effect and the retarding effect due to the Basset force. The paper explores the connection between fractional Brownian motion and anomalous diffusion. The paper also includes an appendix on the Wright function and references to recent developments in the field. The revised version includes corrections, improvements, and updated references, reflecting the author's ongoing research in the area of fractional calculus and its applications.