In Euclidean space, soap films are described by minimal surfaces, surfaces whose mean curvature is zero. In Lorentzian geometry, and in particular in spacetimes described by Einstein's theory of general relativity, there is a Lorentzian analog of minimal surfaces, and the presence of these surfaces signals the development of spacetime singularities and the formation of black holes. We will explore the mathematics connecting these seemingly disparate ideas. No prior knowledge of differential geometry or relativity will be assumed.