  The notion of an existentially closed model is generalised to a property of
geometric morphisms between toposes. We show that important properties of
existentially closed models extend to existentially closed geometric morphisms,
such as the fact that every model admits a homomorphism to an existentially
closed one. Other properties do not generalise: classically, there are two
equivalent definitions of an existentially closed model, but this equivalence
breaks down for the generalised notion. We study the interaction of these two
conditions on the topos-theoretic level, and characterise the classifying topos
of the e.c. geometric morphisms when the conditions coincide.
