The Hyperuniverse Programme, introduced in Arrigoni and Friedman (2013),
fosters the search for new set-theoretic axioms. In this paper, we present the procedure
envisaged by the programme to find new axioms and the conceptual framework behind it.
The procedure comes in several steps. Intrinsically motivated axioms are those statements
which are suggested by the standard concept of set, i.e. the `maximal iterative concept',
and the programme identi fies higher-order statements motivated by the maximal iterative
concept. The satisfaction of these statements (H-axioms) in countable transitive models,
the collection of which constitutes the `hyperuniverse' (H), has remarkable 1st-order
consequences, some of which we review in section 5.