TITLE: What notion captures the 'class' of all classes?
QUESTION [14 upvotes]: In ZFC there is no set that is the set of all sets, for this we introduce the notion of class. But then what is the 'class' of all classes, is it actually a class? Do we apply the same idea again? But then at what stage do we stop? Does this show that classes are not the right notion to go beyond sets, but more of an ad-hoc solution?
Further, within foundational category theory, we have the notion of grothendieck universes, if i recall rightly, this is equivalent to introducing an axiom that an inaccessible cardinal exists. Does this subsume, or is equivalent to the notion of classes?
Finally, what is the formalism that uses classes to extend ZFC, is the NBG?

REPLY [9 votes]: To give a very specific  (quite formalistic, and possibly very wrong - depending on your or even my beliefs) answer to some of your questions: 

In ZFC there is no set that is the set of all sets, for this we introduce the notion of class.  

I don't, because I use ZFC.  Whenever I say "class", I mean "formula".  (Today.  I may change my mind tomorrow.)  You may use NBG, of course. 

But then what is the 'class' of all classes.

No such thing, in ZFC.  (Well, there is the set of all formulas.  But that is not what you mean.)  No such thing in NBG either.   Try KM. 

Do we apply the same idea again? But then at what stage do we stop? 

It depends on what you mean by "we".   I stopped at ZFC.  You may go as far as you want. 
You may even use a type-theoretic approach, in which there are infinitely many levels of this
hierarchy. However, once you have countably many levels, you may ask how many levels there are.  But now the pictures looks somewhat similar to a universe $V_{\delta+\omega}$, which ZFC handles very well. 
I seem to recall that Fraenkel-Bar Hillel-Levy, "Foundations of Set Theory", has an enlightening and more detailed discussion of this topic.