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I. General Theory.- 1 Classical and Rational Formulations of Non-equilibrium Thermodynamics.- 1.1 The General Balance Laws of Continuum Physics.- 1.2 The Law of Balance of Entropy.- 1.3 Classical Irreversible Thermodynamics.- 1.4 Rational Thermodynamics.- Problems.- 2 Extended Irreversible Thermodynamics: Evolution Equations.- 2.1 Heat Conduction.- 2.2 One-component Viscous Fluid.- 2.3 The Generalised Entropy Flux and Entropy Production.- 2.4 Linearized Evolution Equations of the Fluxes.- 2.5 Rational Extended Thermodynamics.- 2.6 Some Comments and Perspectives.- 2.7 Entropy Evolution in an Isolated System: An Illustrative Example.- Problems.- 3 Extended Irreversible Thermodynamics: Non-equilibrium Equations of State.- 3.1 Physical Interpretation of the Non-equilibrium Entropy.- 3.2 Non-equilibrium Equations of State: Temperature.- 3.3 Non-equilibrium Equations of State: Thermodynamic Pressure.- 3.4 Convexity Requirements and Stability.- Problems.- 4 Hamiltonian Formulations.- 4.1 GENERIC Formulation.- 4.2 Reversible and Irreversible Kinematics.- 4.3 Governing Equations of EIT.- Problems.- II. Microscopic Foundations.- 5 The Kinetic Theory of Gases.- 5.1 The Basic Concepts of Kinetic Theory.- 5.2 Non-equilibrium Entropy and the Entropy Flux.- 5.3 Grad' Solution.- 5.4 The Relaxation-Time Approximation.- 5.5 Dilute Non-ideal Gases.- 5.6 Non-linear Transport.- 5.7 Beyond the Thirteen-Moment Approximation: Continued-Fraction Expansions of Transport Coefficients.- Problems.- 6 Fluctuation Theory.- 6.1 Einstein' Formula. Second Moments of Equilibrium Fluctuations.- 6.2 Ideal Gases.- 6.3 Fluctuations and Hydrodynamic Stochastic Noise.- 6.4 The Entropy Flux.- 6.5 Application to a Radiative Gas.- 6.6 Onsager' Relations.- Problems.- 7 Information Theory.- 7.1 Basic Concepts.- 7.2 Ideal Gas Under Heat Flux and Viscous Pressure.- 7.3 Ideal Gas Under Shear Flow: Non-linear Analysis.- 7.4 Ideal Gas Submitted to a Heat Flux: Non-linear Analysis.- 7.5 Relativistic Ideal Gas Under a