Is there a "fundamental problem of thermodynamics"? The "fundamental problem of mechanics" can be boiled down to finding and solving the equation of motion of a system. Similar statements can be said for quantum mechanics for the Schrödinger equation and for electrodynamics and Maxwell's equations, etc. But is there such a thing for thermodynamics? Is there a formulation that allows for this kind of perspective?
 A: For the case of equilibrium thermodynamics, a central task which might be called "the fundamental problem" is to find an expression for a thermodynamic potential in terms of its natural variables. From such an expression all the information about equilibrium states and quasistatic processes can be derived, such as heat capacities, response functions in general, free energy, phase boundaries, latent heats, etc. etc.
A: I'm lumping together statistical mechanics with thermodynamics here, since in my mind they are inextricably linked. I would say the fundamental problem of statistical mechanics is to calculate the maximum entropy distribution for a system plus its environment, given a set of constraints and experimental conditions. Then a typical problem in thermodynamics is to understand how various properties of that distribution (such as the average volume) change as the experimental conditions (such as the pressure) are varied.
However, a difference between statistical mechanics and thermodynamics, and other subjects in physics, is that there is a large distance between the fundamental problem you are solving, and the tools used in specific applications (this point is noted in Baierlein's textbook). For example, in systems where the particle number is variable, typically we actually solve problems using the chemical potential, even though deep down the chemical potential arises as a useful concept because of considerations related to the entropy. Or, you might be interested in minimizing one of the various thermodynamic potentials (depending on what experimental parameters are being varied). You usually don't directly think about maximizing the entropy of the system+environment when solving that kind of problem, even though deep down the reason you minimize these potentials is to maximize the entropy.
A: I'd keep apart:

*

*the fundamental statements, i.e. principles of a theory, and

*subjects, problems of interest, and applications of a theory.

Limiting the discussion to branches of classical physics:

*

*classical mechanics: study of the motion (or the statics) of simple mechanical system, with small number of degrees of freedom, whose evolution can be effectively described as deterministic. Principles:

*

*mass conservation of closed systems;

*either Newton mechanics:

*

*inertia and Galileo relativity;

*second principle of dynamics;

*action-reaction;



*or analytical mechanics:

*

*stationary-action principle





*classical thermodynamics: study of the evolution of complex systems, whose number of degrees of freedom is so high that a deterministic solution is practically infeasible. It deals with variation of total energy of a system, as the result of work and heat exchanged with the external environment, and deals with the non-reversibility of physical processes. You can interpret it as a macroscopic "average" description of the a system, whose molecular details can be investigated by means of statistical mechanics. Principles of thermodynamics:


*thermal equilibrium, about heat exchange between two systems at different temperatures. Fundamental experience: why does the temperature of the hot body decrease and the temperature of cold body increase (broadly speaking, why does heat "flow" - even though nothing flows - from hot to cold bodies)?

*first principle or total energy balance of a closed system. Fundamental experience on the calorimeter: where does work done on a calorimeter go?

*second principle, about non-reversibility. Fundamental experiences: why is there  a natural tendency in the evolution of some systems?

*third principle about absolute temperature.

Thermodynamics along with classical mechanics, and the local equilibrium principle, leads to the mechanics of continuum, the branch of classical physics describing the governing equations of continuous media, like solids (mechanics of solids) and fluids (fluid dynamics), that allows you to approach almost every problem in classical physics not involving electromagnetic phenomena, from the first experiences with a calorimeter, to mixtures, to chemical reactions, to thermodynamic transformation, cycles and heat engines, to the study of heat transfer, to study of statics and dynamics of fluids and so on...


*classical electromagnetism: study of the processes involving electromagnetism. Principles (not all independent):

*

*electric charge conservation: you can't create/destroy electric charge;

*Maxwell's equations: how charges and currents create electromagnetic field, and how the EM field behaves;

*Lorentz force: influence of the magnetic field on charges.



A: I think the best description of what is the "fundamental problem" in Thermodynamics remains the one by H.B. Callen in his textbook (Thermodynamics and an Introduction to Thermostatistics. John Wiley & Sons):

The single, all-encompassing problem of thermodynamics is the determination of the equilibrium state that eventually results after removing internal constraints in a closed composite system.

I would add that the system should not necessarily be closed. Even isothermal or isobaric conditions are allowed.
Notice that  Equilibrium Thermodynamics solves the problem by showing that a few special functions of the state variables, the so-called fundamental equations (either the thermodynamic potentials or Massieu's functions), allow the solution of the fundamental problem under different external conditions via maximum or minimum principles.
Another observation is that such a problem is related to but different from the central problem of Equilibrium Statistical Mechanics, which is the determination of the fundamental equation from the knowledge of the microscopic interactions. However, a clear boundary between the two approaches is that within Equilibrium Thermodynamics, all the microscopic degrees of freedom have already been used and eliminated from a description in terms of the few macroscopic variables necessary to describe a system at equilibrium.
Finally, let's note that an essential part of the fundamental problem of thermodynamics, according to Callen's definition, is the identification of the correct set of state variables for each class of thermodynamic systems. Therefore, it remains a problem to be solved every time one would like to treat a new class of systems with the methods of Equilibrium Thermodynamics.
A: As someone already pointed out the fundamental problem of thermodynamics is best described by Herbert Callen in his book. He essentially reduces the principle of thermodynamics in terms of postulates of physically relevant variables in the system.

The first postulate states 
Postulate 1: There exist particular states (called equilibrium states) of simple systems that, macroscopically, are characterized completely by the internal energy $U$, the volume $V$, and the mole numbers $N_1$, $N_2$..., $N_r$ of the chemical components.
The second postulate involves the definition of a function which is useful for describing thermodynamic equilibrium in terms of it.
Postulate 2: There exists a function (called the entropy $S$) of the extensive parameters of any composite system, defined for all equilibrium states and having the following property: The values assumed by the extensive parameters in the absence of an internal constraint are those that maximize the entropy over the manifold of constrained equilibrium states.
This essentially states that the problem of thermodynamic equilibrium is reduced the problem maximizing the entropy function of the system, subject to constraints of the system.
The other two postulates are considered to provide a formal definition of the absolute temperature scale (which will be non-negative) and the measure of entropy going to zero at absolute zero Kelvin.
Postulate 3: The entropy of a composite system is additive over the constituent subsystems. The entropy is continuous and differentiable and is a monotonically increasing function of the energy.
This ensures two things, primarily, entropy is first-order homogeneous function of extensive parameters, $U, V, N$ and secondly it also ensures $\big(\frac{\partial S}{\partial U}\big)_{N,V} > 0$, which implies non-negative temperature, T being defined as $T = \big(\frac{\partial U}{\partial S}\big)_{N,V}$.
Postulate 4:Entropy of the system vanishes for any state at T = 0.
With these four postulates, Callen gives a detailed exposition of how one can derive all the thermodynamic relationships with the help of some fundamental relationships like energy and mass conservation.
I will present here a brief example of describing thermal and mechanical equilibrium.
Consider an isolated system composed of two subsystems (1&2) with energies and volumes $U_1, V_1$ and $U_2, V_2$ respectively. They separated by an adiabatic wall which is free to move, allowing exchange heat and adjust their volumes to reach both thermal and mechanical equilibrium. With the additivity from Postulate 3, the entropy of the total system can be given as,
$$
S = S_1(U_1, V_1, N_1) + S_2(U_2, V_2, N_2)
$$
As the energies and volumes of the system are changing we can write the entropy maximization (extremum here) as,
$$
dS = 
\bigg(\frac{\partial S}{\partial U_1}\bigg)_{N_1,V_1}dU_1 +  \bigg(\frac{\partial S}{\partial U_2}\bigg)_{N_2,V_2}dU_2 +  \bigg(\frac{\partial S}{\partial V_1}\bigg)_{N_1,U_1}dV_1 + \bigg(\frac{\partial S}{\partial V_2}\bigg)_{N_2,U_2}dV_2 = 0
$$
Since the total system is isolated, the changes in their energies and volumes are related as $dU_1 = -dU_2$ and $dV_1 = -dV_2$.
Using these relationships and definitions of pressure and temperature in terms of entropy function,
$$
\bigg(\frac{1}{T_1} - \frac{1}{T_2}\bigg) = 0 \\
\bigg(\frac{P_1}{T_1} - \frac{P_2}{T_2}\bigg) = 0 \\
$$
This equation essentially describes that in the thermal and mechanical equilibrium, the energies and volumes of the subsystems 1&2 adjust themselves such that the temperatures and pressures of the subsystems are equal.
It is crucial to note that we considered only the first derivative of entropy $dS$, which amounts to calculating to the extremum (both maximum & minimum). In particular the condition that describes the equilibrium has to mathematically be maximizing entropy that corresponds to,
$$
\bigg(\frac{\partial^2 S}{\partial^2 U}\bigg)_{N,V} \le 0 \\
\bigg(\frac{\partial^2 S}{\partial^2 V}\bigg)_{N,U} \le 0 
$$
These conditions prescribe the physical behaviour of various response functions like specific heat $C_V$ and compressibility $\kappa_T$ and so on. They essentially dictate that these functions also need to positive in order for a thermodynamic system.
Ref. Herbert B Callen, Thermodynamics and an introduction to thermostatistics
PS: An equivalent formalism of the thermodynamic equilibrium can also obtained from the minimization of energies subject to different constraints, which lead to the definition of different types of free energies.
