The usual ‘closed-graph theorems’ are concerned with the question of which topological structures on sets X, Y will ensure equivalence of continuity and closed-graph conditions for certain mappings t: X → Y. In this paper some stronger closed-graph like conditions are introduced on a mapping between uniform spaces, by considering the ‘hypergraph’, or graph of the induced mapping between hyperspaces. The central result, included in Theorem 1, states that for arbitrary uniform spaces X, Y, t is continuous if its hypergraph is closed. Thus for topological vector spaces (and some others) the closed-hypergraph condition is equivalent to continuity. In section 2 some situations are found in which continuity is implied by certain intermediate conditions to closed-graph and closed-hypergraph conditions. In particular, a closed-hypergraph theorem is shown to hold for locally convex spaces, provided only that X is barrelled. Finally, in section 3, the hypergraph of a relation is studied, and the separatedness of the quotient space of an equivalence relation is shown to replace continuity with regard to closed-graph and closed-hypergraph conditions.