Is Newton's third law always correct? Newton's third law states that every force has an equal and opposite reaction. But this doesn't seem like the case in the following scenario:
For example, a person punches a wall and the wall breaks.  The wall wasn't able to withstand the force, nor provide equal force in opposite direction to stop the punch.
If the force was indeed equal, wouldn't the punch not break the wall? I.e., like punching concrete, you'll just hurt your hand. Doesn't this mean Newton's third law is wrong in these cases?
 A: If the wall breaks, that just implies that it was not strong enough to resist the force of the push you tried to apply. It also means that you did not manage to apply the full force, as the wall broke before you reached that level.
The above should be slightly modified to take into account static vs dynamic friction. Static friction (without movement) is higher than the dynamic friction when the object first starts to move. This concept is familiar to anyone who has tried braking on a slippery road. With low brake force, the tyres keep rolling, and the contact point with the road does not move with respect to the road. If the brake force exceed the maximum friction the tyres can provide then the wheels lock up, and it suddenly feels as if the car shoots forward. At that point it is better to release the brakes and try again. Modern cars with ABS systems do this automatically, many times per second, and you can feel it as a juddering during an emergency stop.
The same can be true for the wall: the force to break the wall may be stronger than that required to push the pieces further apart.
A: The wall cannot react completely by stopping the blow, but this is not the only way that it can react. It will transfer as much of the blow's energy back to your hand as possible, which is why your hand hurts. But once the wall has reached its limits there, it has to drain off the excess in other ways. Many objects can do this by moving -draining off the energy of the blow as kinetic energy- which is part of how the famous demonstration with the hanging metal balls works. But the wall is anchored in place, so it can't move. Objects can also deform to at least some degree, but the wall is likely made of material that cannot deform very much, so while a little energy still goes into deforming the wall, that can only go so far.
Another possibility is to break apart. Once this happens, at least some of the pieces are no longer anchored in place, and a lot of the energy of the blow can be transferred to them as kinetic energy. This is why small pieces of the wall go flying all over the place instead of staying in a neat pile near the remains of the wall.
Some energy also gets released to the air, first as the sound of the blow hitting the wall, and then as the sound of the wall breaking. Realistically speaking, some of the energy will also be released as heat. None of these is going to be a huge factor, compared to what gets transferred back into your arm or into the wall's destruction. But these remaining factors, and others, help to account for the energy that escapes the wall-fist system. Eventually, it all adds back up.
Another thing to consider is that once the wall breaks, it's no longer in the blow's path, which prevents the blow from applying any further force. This isn't so different from what happens when the object moves without breaking.
Lastly, as other answers have pointed out, Newton's Third Law is not universal. It's actually more of a special case. It happens to be a very large special case -it works well enough in our particular corner of the universe to be useful in everyday life situations- but there do exist cases where it doesn't hold. This just isn't one of them.
A: Despite 11 answers to this question already, I don't feel that any have answered the question well.
(Note: This answer is simplified and assumes the punch is slow enough to ignore inertia and relativity)
Firstly, let's look at force at the atomic level. This is where the force is really happening. The forces that we feel in everyday life are generally the forces between atoms and molecules (intermolecular forces). I'll use Helium atoms as an example, because they're easy to draw. When two He atoms get close together, their electron shells overlap and cause them to repel each other. Note that you never get a situation where one atom repels, and the other does nothing, or one repels and one attracts. Always they both repel each other, or both attract each other, and both atoms feel the same magnitude force, in exactly opposite directions.

The force they feel is a function of the distance between them. The force between them behaves basically like a spring. In the illustration above, the two atoms are repelling each other, and will accelerate away from each other. As they move apart, the force decreases, until at a certain point, it reaches zero, and we consider them not to be 'touching' any more.
Now imagine we start with one atom stationary, and throw another atom at it. When the moving atom gets close enough to the stationary one, they will feel the force of repulsion. Both will accelerate based on the force between them. They accelerate in opposite directions, so the stationary atom accelerates and flies off, while the moving one decelerates to a stop.
Molecules behave in a similar way towards each other.
Since a wall is made up of molecules, it behaves pretty much like the force between molecules, except in a solid object, neighboring molecules are bonded together, meaning that when you push them closer together, they repel, and when you pull them further apart, they attract. The wall is basically a very stiff spring. When you push on a wall, it bends.

Bending is the only way it can push back on you. Bending means that some of the molecules in the wall are pushed closer together, and some are pulled further apart. The harder you push, the more it bends. It bends just so that it's pushing back on you as hard as you're pushing. If you're pushing with a constant force, everything is in equilibrium, and all the force vectors acting on each molecule add up to zero, so nothing is accelerating.
If you push hard enough, you'll manage to stretch some molecules far enough apart that their bond breaks. At that point the force between them drops to zero. Now those molecules are not in equilibrium, and they will accelerate away from each other.
If you push hard enough, and the wall breaks, it's no longer bending, it's accelerating away from your hand, just like the atoms in the example above. As it accelerates away, the force between your hand and the wall decreases and reaches zero when your hand and the wall are no longer 'touching'.

When you punch a wall, the forces you and the wall are feeling are entirely made up of the forces between atoms and molecules. So whether the wall stands or falls, Newton's 3rd law holds the whole time. The wall can only push back on your hand to the extent that it can bend without  breaking.
But what if I push really hard on the wall?
The answer is you can't. You can put a lot of effort into the punch, but if you were to measure the actual force applied to the wall, it would increase up to the point, then the wall would break, then the force would drop back down to zero.
Newton's 3rd law doesn't mean that everything is indestructible.

Added:
If you haven't already discovered Veritasium's excellent YouTube channel, you should. He has a good video helping us to understand Newton's Third Law:

A: Nice question. It's a common confusion among many beginning students. When I push something, shouldn't it stay still as there's an equal and opposite reaction to counterbalance my force?
The Answer: The two forces in question act on two different bodies.
The resistance force of wall has nothing to do with its equal and opposite reaction. The reaction is acting on hand, not the wall itself to prevent its own motion.
A: No, it does not mean that Newton's 3rd law is not correct. The wall pushed back (your hand hurts), but the force you applied broke the wall and pushed pieces forward. I will try to list the forces.
Hand pushes wall - wall pushes back
hand moves wall - wall resists moving
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ - sound is made
Did I miss anything?
A: When a body exerts a force on another body, both being in contact, the other body also exerts an equal and opposite force at their contact. 
The first body may push the second and make it move, which is because of a non-zero net external force. But the force the two bodies are exerting on each other (internally) at their contact are equal and opposite. The first block is able to move the second block because the external force one first block is greater than that on the second block. 
The internal forces inside a body cancel out from Newton's third law.
A: You can't apply to a wall the force bigger than it can withstand. If the wall breaks at some certain amount of newtons of the force applied then it follows that you applied this certain force. And of course the reaction of the wall equals the force applied.
A: Almost off-topic, it's worth mentioning that Newton's laws only apply in a Galilean reference frame, which is rather utopic (making Newton's law an approximation of the reality… but what else is Physics anyway?)
In any case, other answers were right: the wall has a reaction from you (and it may break) and also applies a force on you (you may feel pain in your hand).
A: Newton's third law is not always correct, contrary to what you may have heard. It is correct in the context of newtonian mechanics, because we assume then that point particles are described only by their mass, and symmetry and conservation of momentum of the system imply that the third law must apply for the case of a closed system, which the universe is defined to be. It does not apply in the case of macroscopic bodies in general, because they deform themselves, as you described; energy is dissipated, hence the system is not closed. Also, it does not apply in the case of electromagnetism, because the field carries part of the momentum of the system.
A: (As suggested in Classical Mechanics by H. Goldstein, 3rd edition in chapter 1 : Survey of the Elementary Particles )
No, the third famous law is not always valid. As pointed out above, in the case of electromagnetism, take for an example, two charged particles A and B are in motion.
B is just travelling perpendicular to the path of A and is right on the axis of A's motion.
You can calculate Coulomb's force for one due to another. But try finding the magnetic force due to one on the other. You will find the Lorentz force (sum of electric and magnetic forces)  on one is not equal to the other. 
Voila! Newton's third law violated!!
Well, if the fields concept is taken into account, and it has to be, then the third law is improved and protected: no violation. But excluding that, we can say it is not a strong law.
A: 
For example, a person punches a wall and the wall breaks. The wall wasn't able to withstand the force, nor provide equal force in opposite direction to stop the punch.

Two separate events happened here.
First, your muscles accelerated your fist. The equal and opposite reaction was the rest of your body moving backwards. Since your body weighs a lot more than your arm it moves backward a lesser amount. 
Now your fist has kinetic energy. When this is delivered to the wall, the wall will move away based on the mass x velocity of your fist vs. the inertia of the wall. Factor in the crush/breakability of the wall (and your fist) and we will eventually reach equilibrium. The debris (and/or blood) on the floor represents the expended energy of the event, but if you add everything up (and I do mean everything )it will all be centered in the same place as before.
A: 
For example, a person punches a wall and the wall breaks. The wall wasn't able to withstand the force, nor provide equal force in opposite direction to stop the punch.
If the force was indeed equal, wouldn't the punch not break the wall? I.e., like punching concrete, you'll just hurt your hand. Doesn't this mean Newton's third law is wrong in these cases?

Nothing says that walls must provide enough force to stop a punch. In this case, the person's fist will carry on moving, through the space which the wall used to occupy, until some other object exerts a force on it.
For example, once the person's arm becomes fully extended and resists being stretched, that may stop their fist. Alternatively, if the punch was really strong, their whole body may get dragged along (and the Earth will accelerate a little in the opposite direction, like when someone jumps).
The fist will be slowed down, in direct proportion to the amount of force needed to break the wall, since force = mass * acceleration (remember that slowing down is still an acceleration!)
A: I think that this third law is all about Conservation of Energy. When a person punches a wall and the wall breaks; some of the energy is transfered to the wall and the wall breaks because of not having strength to the punch force. Some of the energy turns to heat and internal energy, the effects of which is not that much obvious to the person, and some of the energy turns back to the person's body and turns to his internal energy which he feels some pain in his hand. Imagine if the person breaks his hand while punching the wall. That is because his bones do not have strength to dissipate the energy which comes back to his body. All is the matter of energy.
A: The reason it appears that Newton's 3rd law does not apply in your example, is that there is some confusion and wrong assumptions going on.  It should be obvious that a wall can only provide a "reactive" force up to its "breaking limit," which depends on its dimensions and composition. Obviously, a thin wall made of sheet-rock will not "resist" as much as a thick wall made of steel! There is also an assumption that the wall is somehow "held in place." Otherwise, if the wall is allowed to move, the reactive force will be limited to the time the fist is in contact with the wall. In other words, if the wall is strong enough to resist the force (and immobile), it will provide a reactive force equal and opposite to the applied force. 
