Really how does the entropy of the universe increase? Universe means the system along with its surroundings. I have always got this statement while studying the second law; be it a thermodynamics book (Sears, Salinger) , physical chemistry book (Atkins, Paula) or any site. But really how does this happen? 
Take for example, the isothermal expansion of gas; the increase in entropy of the gas(which is our system) is given as $$\Delta S_{\text{sys}} = nR\ln\dfrac{v_f}{v_i}$$ 
 Since the the surroundings remain at constant pressure, the change in enthalpy is same as the heat energy taken by the system. Therefore the entropy of the surroundings is given by $$\Delta S_\text{surr} = -nR\ln\dfrac{v_f}{v_i}$$
 Therefore the entropy if the universe is what? $0$; Then how can the entropy of the universe increase? It remains the same! Then why is the statement telling otherwise?
 A: 
Really how does the entropy of the universe increase?

The statement "entropy of the universe increases" is a doubtful extrapolation of thermodynamics.
The problem with that statement, as CuriousOne has written in the comments, is that universe is not a closed system amenable to thermodynamic description. It has no volume, no temperature and we have no means to experiment with it the way we do with thermodynamic system such as gas in a piston. For example, heat exchange between the Universe and environment has no meaning because there is no environment.
Thus thermodynamic laws do not apply to it with the same certainty they do to lab systems and it has no meaning to ascribe it the Clausius entropy (via integral of $dQ/T$).
To make the statements in books meaningful, often it helps to replace the word "universe" with "closed system consisting of the studied system and the surrounding system it interacts with" supposing these two can attain states of thermodynamic equilibrium. Then the entropy or any other thermodynamic state quantity ascribed to these may have meaning.
A: 
Then how can the entropy of the universe increase?

Because the universe is like your gas with no surroundings. But see the stress–energy–momentum tensor, and note that it "describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics". See the shear stress term? Space is more like a gin-clear ghostly elastic solid than a gas. And note the energy-pressure diagonal? Space has this innate "pressure". If you've got one, squeeze a stress-ball down in your fist, then let go. Watch it expand. 
Oh, and note that there is no actual evidence for "the impossibility of boundaries". I think that's just a failure of imagination myself. In the old days some people couldn't conceive of a world that didn't have an edge. Nowadays some people can't conceive of a world that does.  
A: A quick answer is that the entropy of the universe is actually encoded in the Weyl curvature. Einstein's field equations:
$R_{ab} - \frac{1}{2}g_{ab} R + \Lambda g_{ab} = \kappa T_{ab}$
contain only ten independent components of the Ricci tensor for a 4-D spacetime. However, this is obtained from a contraction of the Riemann curvature tensor, which itself for a 4-D spacetime has 20 independent components. So, from a cosmological perspective, the Einstein equations are not enough to understand the evolution of the universe, one must see where the missing 10 independent components are, and those are in the Weyl tensor:
$C_{abcd}=R_{abcd}-\frac{2}{n-2}(g_{a[c}R_{d]b}-g_{b[c}R_{d]a})+\frac{2}{(n-1)(n-2)}R~g_{a[c}g_{d]b}$
The point is that at the big bang, the Weyl tensor vanishes, and as the universe expands, (and matter is created, and so forth), the Weyl tensor components become larger. This is a mathematically plausible explanation of why the entropy of the universe increases, since, you cannot use standard old thermodynamics, as they do not include gravity. Once you include gravity and Einstein's equations, the story changes, and Penrose's Weyl Curvature hypothesis seems to be the only reasonable explanation that doesn't involve invoking "exotic physics". Penrose's paper for reference is: "R. Penrose. Singularities and time-asymmetry. In General relativity: an Einstein centenary survey, volume 1,
pages 581–638, 1979"
A: When energy flows from a hotter subsystem to a colder subsystem the entropy of the combined system increases.
$$\frac{dS_1}{dE_1}=\frac{1}{k_BT_1}<\frac{1}{k_BT_2}=\frac{dS_2}{dE_2}.$$
So if they exchange energy in a way that conserves energy then we get $-\Delta S_1<\Delta S_2$ and the total entropy increases.
This does require that the systems have a temperature. If they don't you can try breaking it into regions that each have a temperature and see if you can apply the same idea locally.
This requires that energy is conserved. Energy is frame dependent so you might need a global frame so each subsystem can have an energy. This doesn't always happen. And then you want energy to be conserved between the two, this also doesn't always happen even if you pick some frame. General Relativity would be a common culprate.
And to get the total entropy to increase we assumed the entropy of the system was the sum of the entropies of the parts. This is not always true it comes from the assumption that allowed states for the system are products of states of the subsystem and if they aren't independent enough this isn't always true.
But it becomes hard to call them separate subsystems on those cases. But it was a difficult line anyway since we want them to interact.
You could instead model the entire system as a whole and make new ways to study it based on assigning properties of the subparts. But this is then a different subject matter. Plus an objective description of the whole can be very different than the subjective descriptions parts make about other parts so it might not be wise to use the same words.
However, I think the above equation is intended to explain why energy spontaneously flows from bother to colder (even when that makes the colder thing colder which happens regularly in gravitationally bound subsystems) and that it does so like an entropic force, the energy has many more interactions that increase the entropy than don't and that in the situations where there are temperatures defined this results in heat going from hotter to colder.
So I think most people take the entropic force as primitive and derive other things (such as heat flow) from it. Not the other way around.
