There are various approaches to having classes as formal objects in set theory, the two most common being Gödel-Bernays set theory and Kelly-Morse set theory. In both of these theories, one has several ways to think about classes of classes.

On the one hand, one can have a class of classes in the sense that there is a class $U\subset V\times V$ of pairs, and one thinks of this as an indexed family of classes $U_a=\{b\mid (a,b)\in U\}$. Thus, one thinks of a classes of classes as a subset of the plane, with the classes in this meta class being the slices of that subset. For this notion of classes of classes, there can be no class of all classes, since we can form the class $D=\{a\mid a\notin U_a\}$, which by the usual diagonal argument cannot occur as a slice in $U$.

On the other hand, one can consider in GB and KM set theory the meta-classes of definable collections of classes, much like one considers definable collections of sets as classes in ZFC. For any formula $\varphi(X)$ in the language with a class parameter $X$, one may consider the meta-class of all classes for which $\varphi(X)$. If this meta-class contains any proper classes, then it cannot itself literally be itself a class, since every class has only sets as members. So although we can speak of the meta-class of all classes $X$ such that $X=X$, say, which would be the meta-class of all classes, this meta-class is not a class.

Meanwhile, there are various set theories that allow the construction of sets to continue far past what would otherwise be a perfectly acceptable universe of sets. For example, the Grothendieck universe concept is like this, or $H_{\kappa}$ for $\kappa$ inaccessible or even merely a worldly cardinal. Ackerman set theory also has this feature.
For none of these theories is there a class of all classes, by essentially the same diagonal argument.

(Meanwhile, in Quine's New Foundations, there *is* a set of all sets, and the usual set-class distinction is less present.)