Trapped surfaces are a central topic of study in mathematical general relativity. Penrose's incompleteness theorem (1965) tells us that the presence of these surfaces in a suitable class of spacetimes implies that the spacetime is geodesically incomplete, thus tying the concept of trapped surfaces to the study of incompleteness and singularity formation in GR. In this talk, we will introduce the concept of trapped surfaces and discuss Christodoulou's breakthrough result from 2009 showing that closed trapped surfaces can form in vacuum via the focusing of incoming gravitational radiation. Speaker(s): Chris Stith (University of Michigan)