The paradox of irreversibility raised by the kinetic theory of gases has constantly animated the scientific community. Classical mechanics describes the motion of bodies such as atoms and molecules by means of time reversible equations. By contrast, thermodynamics emphasizes the irreversible character of the macroscopic phenomena such as viscosity. Whereas Boltzmann proposed a probabilistic explanation, recent advances show that the hypothesis of microscopic chaos plays a key role and provides a dynamical interpretation of the emergence of irreversibility. In this book, using viscosity as a leading concept, we sketch the historical development of the concepts related to this fundamental issue. A new method is presented and used to calculate the viscosity by molecular dynamics in periodic systems. This is applied in the context of new theories that establish relationships between viscosity and properties of the microscopic chaos. As a result, this book should be useful for students and scientists looking to become familiar with recent advances in statistical mechanics embedded in the long history of the areas of physics.