In logic (especially in model theory and set theory), one defines a model $\mathcal A$ for a given signature $S$ to be given by a underlying set $A$ and some additional structure (by which I mean: for every constant symbol in $S$ an element of $A$, for every $n$-ary function symbol in $S$ a function $A^n\to A$, and so on). In set theory, one often considers models satisfying some set theory axioms (often one takes ZFC); these models have the form $(V, \in)$, where $V$ is called a universe and $\in$ a binary relation on $V$.
I find it philosophically unsatisfying that set-theoretic universes should be sets, since within a set-theoretic universe satisfying ZFC, this universe thinks that it is a proper class, if you know what I mean.
Why does one not allow models/universes to be proper classes?
Why are the set-theoretic universes sets?