What are the contributions of the Non-equilibrium Thermodynamics? I am very interested in Thermodynamics, but though it is easy to find good books and information about classical thermo, it seems that is not the same for non-equilibrium thermodynamics. I would like to know what important discoveries (theoretical or practical) has the non-equilibrium thermo has made, and what things of this discipline are really worth knowing. 
 A: Non-equilibrium thermodynamics is an emerging field of study. Asking whether the new concepts introduced by that emerging field will prove to be of practical value is a bit premature. The concepts are only now just starting to make their way out of academia.
The concept certainly has merit. On a grand scale, a thermodynamic system that truly is in equilibrium with the external environment necessarily has a temperature of 2.72548 K (the temperature of the cosmic microwave background). All thermodynamic systems of interest to humankind are far removed from thermodynamic equilibrium.
On the other hand, does non-equilibrium thermodynamics offer anything above the simple expedient of making intrinsic quantities such as temperature, pressure, density, etc. local functions that vary within some object as opposed to global attributes of that object? Nobody teaches those simple expedients as "non-equilibrium thermodynamics"; they're part and parcel of standard thermodynamics. But if you think about it, that one has to resort to these expedients inherently means the system is not in thermodynamic equilibrium.
A: I agree with the last part of David Hammen's answer that the discrimination between classical thermodynamics and non-equilibrium thermodynamics is sometimes a bit arbitrary and essentially a cultural trend. There is in fact nothing conceptually new in the quantities introduced: temperatures are still temperatures and so are pressures, densities and so on and the fundamental principles to be fulfilled are still the first and second principles of standard thermodynamics. 
Now, I am being a bit unfair here as non-equilibrium thermodynamics worries about something that classical thermodynamics did not worry explicitly about that is the "dynamics" of thermodynamical systems.
Standard thermodynamics provides rules for picking transformations that may occur among all the possible transformations one might think of. However, it is quite silent on the subject of how long does it take to reach such and such states.
Empirical laws, which are the dynamical equivalents of the equilibrium equation of state, may be inferred from experiments (like Fick's laws and Fourier's Laws) but finding a theoretical rationale for these laws and their range of validity is one of the goals of most non-equilibrium schools of thermodynamics.
Now, if I have to name tools and theories that have been used to address this dynamical problem, I would mention (this is not an exhaustive list) Swanzig's projection method, large deviation theory, generalized fluctuation-dissipation theorems, stochastic processes and simply the linear response theory which is assumed anyway in many cases.
