This talk will focus on two distinct measures of singularities: test ideals (in positive characteristic) and multiplier ideals (characteristic zero). Though known for over a decade to be related via reduction to characteristic p > 0, recent advances have provided a uniform description of these invariants using regular alterations. This description, which shall be presented in detail, simultaneously sheds new light on both the connection and differences between test and multiplier ideals. Parts of the talk are based on joint works with Manuel Blickle, Karl Schwede, and Wenliang Zhang.