Probably it is not a very common terminology, but I'm used to the following natural definitions:

Definition 1. Let $\mathcal{A}$ and $\mathcal{B}$ be categories. The category $\mathcal{A}$ is called $\mathcal{B}$-complete ($\mathcal{B}$-cocomplete) iff for every functor $\mathcal{F}\colon\mathcal{B}\to\mathcal{A}$ there exists a projective (inductive) limit of $\mathcal{F}$.

Definition 2. Let $\mathcal{U}$ be a Grothendieck universe, $\mathcal{A}$ be a category. The category $\mathcal{A}$ is called $\mathcal{U}$-complete ($\mathcal{U}$-cocomplete) iff it is $\mathcal{B}$-complete ($\mathcal{B}$-cocomplete) for every $\mathcal{U}$-small category $\mathcal{B}$.

(Note, that the definitions 1 and 2 don't disagree with each other, because from the set-theoretic point of view, a universe cannot be a category).

Thus the framework of universes allows us to consider different types of small-completeness. All results concerning limits/completeness presented in the class-theoretic framework can easily be transferred (and even generalized) to the framework of universes. As an example, every $\mathcal{U}$-small $\mathcal{U}$-complete category is a complete preorder. By the Grothendieck's axiom, every category is $\mathcal{U}$-small for some universe $\mathcal{U}$, therefore categories which have all limits are not very interesting (they are preorders).

Another picture arises when we talk about the preservation (reflection etc) of limits (colimits). For instance, every right adjoint functor preserves all limits (is $\mathcal{U}$-continuous for every Grothendieck universe $\mathcal{U}$). Another example: fully faithful functors reflect all limits. It is worth to note, that these statements are invariant under changes of foundations (universes or classes).

Introduction au langage fonctoriel), where this is treated in detail (Section I.6). $\endgroup$ – Fred Rohrer Nov 17 '18 at 11:56