From Kinetic Models to Hydrodynamics : Some Novel Results 1. Introduction 2. From the Phase Space to the Boltzmann Equation 3. Methods of Reduced Description 4. Hydrodynamic Spectrum of Simple Fluids 5. Hydrodynamic Fluctuations from the Boltzmann Equation 6. 13 Moment Grad System 7. Conclusions References. . From Kinetic Models to Hydrodynamics serves as an introduction to the asymptotic methods necessary to obtain hydrodynamic equations from a fundamental description using kinetic theory models and the Boltzmann equation. The work is a survey of an active research area which aims to bridge time and length scales from the particle like description inherent in Boltzmann equation theory to a fully established “continuum” approach typical of macroscopic laws of physics.The author sheds light on a new method—using invariant manifolds—which addresses a functional equation for the nonequilibrium single particle distribution function. This method allows one to find exact and thermodynamically consistent expressions for: hydrodynamic modes; transport coefficient expressions for hydrodynamic modes; and transport coefficients of a fluid beyond the traditional hydrodynamic limit. The invariant manifold method paves the way to establish a needed bridge between Boltzmann equation theory and a particle based theory of hydrodynamics. Finally the author explores the ambitious and longstanding task of obtaining hydrodynamic constitutive equations from their kinetic counterparts. The work is intended for specialists in kinetic theory—or more generally statistical mechanics—and will provide a bridge between a physical and mathematical approach to solve real world problems.