  The Heat theorem reveals the second law of equilibrium Thermodynamics
(i.e.existence of Entropy) as a manifestation of a general property of
Hamiltonian Mechanics and of the Ergodic Hypothesis, valid for 1 as well as
$10^{23}$ degrees of freedom systems, {\it i.e.} for simple as well as very
complex systems, and reflecting the Hamiltonian nature of the microscopic
motion. In Nonequilibrium Thermodynamics theorems of comparable generality do
not seem to be available. Yet it is possible to find general, model
independent, properties valid even for simple chaotic systems ({\it i.e.} the
hyperbolic ones), which acquire special interest for large systems: the Chaotic
Hypothesis leads to the Fluctuation Theorem which provides general properties
of certain very large fluctuations and reflects the time-reversal symmetry.
Implications on Fluids and Quantum systems are briefly hinted. The physical
meaning of the Chaotic Hypothesis, of SRB distributions and of the Fluctuation
Theorem is discussed in the context of their interpretation and relevance in
terms of Coarse Grained Partitions of phase space. This review is written
taking some care that each section and appendix is readable either
independently of the rest or with only few cross references.
