The problem of finding proper set-theoretic foundations forcategory theory has challenged mathematician since the very beginning. In this paper we give an analysis of some of the standard approaches that havebeen proposed in the past 70 years. By means of the central notions of class and universe we suggest a possible conceptual recasting of these proposals. We focus on the intended semantics for the notion of large category in each proposed foundation. Following Feferman we give a comparison and evaluation of (...) their expressive power. (shrink)