The (maximal) iterative concept of set is standardly taken to justify ZFC and some of its extensions. In this paper, we show that the maximal iterative concept also lies behind a class of further maximality principles expressing the maximality of the universe of sets V in height and width. These principles have been heavily investigated by the first author and his collaborators within the Hyperuniverse Programme. The programme is based on two essential tools: the class="EmphasisTypeItalic ">hyperuniverse, consisting of all countable transitive models of ZFC, and V -logic, both of which are also fully discussed in the paper.