We present a theorem of Foreman that allows an exact characterization of what happens to the structure of precipitous ideals after suitable forcing. This theorem unifies several well-known results, giving as them quick corollaries. We will use it to show: forcing precipitous ideals from large cardinals, preservation theorems of Kakuda and Baumgartner-Taylor, and Solovay's consistency result on real-valued measurable cardinals.  We will also show some new applications due to the speaker.