The general type of problem considered here is the following. Suppose I I is a countably complete ideal on ω 1 {\omega _1} satisfying some fairly strong saturation requirement (e.g. I I is precipitous or ω 2 {\omega _2} -saturated), and suppose that P P is a partial ordering satisfying some kind of chain condition requirement (e.g. P P has the c.c.c. or forcing with P P preserves ω 1 {\omega _1} ). Does it follow that forcing with P P preserves the saturation property of I I ? In this context we consider not only precipitous and ω 2 {\omega _2} -saturated ideals, but we also introduce and study a class of ideals that are characterized by a property lying strictly between these two notions. Some generalized versions of Chang’s conjecture and Kurepa’s hypothesis also arise naturally from these considerations.