This volume presents an original, synthesizing approach to the thermal physics of fluid continua based on a novel extension of Hamilton's principle which allows the free flow of entropy, independent of that of matter. The extension, used in the context of NAther's theorem of variational calculus, gives rise to heat and diffusion terms in the conservation laws for the energy, momentum and matter, and may include the effects of thermal inertia. The mass conservation in reacting systems results from their intrinsic symmetries. The role of thermodynamic irreversibility can be investigated through a generalized action functional with Onsager's dissipation potentials. While variational calculus is the basic mathematical tool, the book emphasizes the conservation laws in the context of the underlying thermodynamics (reversible or not) rather than the mathematical formalism. The book can be used as a supplementary text in graduate courses on fluid mechanics, nonequilibrium thermodynamics, transport phenomena and variational calculus. As a reference text for further research it will attract researchers working in various branches of macroscopic physics/chemistry and applied mathematics, especially those in continuum mechanics, nonequilibrium thermodynamics (classical and extended), heat and mass transfer, etc. Applied mathematicians will welcome the use of the field (Lagrangian and Hamiltonian) formalisms for complex physiochemical continua.