  We study ideals in, and continuity of, quantaloid-enriched categories
(Q-categories for short) as a 'many-valued and many-typed' generalization of
domain theory. Abstractly, for any (saturated) class Phi of presheaves, we
define and study the Phi-continuity of Q-categories. Concretely, we compute
three examples of such saturated classes of presheaves - the class of flat
ideals, the class of irreducible ideals and the class of conical ideals - which
are proper generalizations of ideals in domain theory.
