f-singularities are classes of singularities defined by the behavior of frobenius. a prominent tool for measuing these singularities is the test ideal, a characteristic $p > 0$ analog of the multiplier ideal. recently, there has been interest in applying the methods of f-singularities to a number of geometric problems in positive characteristic. however, one gap in the theory has been the behavior of f-singularities in families. for example generic restriction theorems for test ideals have been lacking. in this talk, i will discuss recent joint work with zsolt patakfalvi and wenliang zhang on the behavior of f-singularities and test ideals in families. i will first define the relevant terms and explain why you these definitions and methods are useful (as a replacement for kodaira vanishing in characterisic $p > 0$). we will then obtain generic (and non-generic) restrictions theorems for test ideals. some global geometric consequences will also be discussed if there is sufficient time.