The last forty years has seen increasingly interesting differential geometry problems inspired by physical motivations from general relativity. For example, the fact that the 3-torus does not admit a metric of positive scalar curvature (a beautiful geometric theorem that improves one's understanding of scalar curvature) can be used to prove the positive mass theorem of general relativity (which says that positive mass objects collectively have a positive total mass). The links between these natural physical questions and these interesting geometric questions will be described in the minicourse, and the proofs of many of the geometric theorems will be given. In particular, we will discuss the relationship between minimal surfaces (surfaces with zero mean curvature which are at critical points of the area functional) and black holes, the proof of the positive mass theorem, the proofs of the Riemannian Penrose inequality (which is motivated by the physical statement that the total mass of a spacetime including only positive masses and some black holes should be at least the mass of the black holes), and the techniques used to prove these theorems. This will lead us to discuss geometric flow techniques, including inverse mean curvature flow as well as the conformal flow of metrics used by the instructor to prove the Riemannian Penrose inequality. If there is time we may also discuss how some of these techniques relate to the geometrization of 3-manifolds by means of Yamabe invariants and Ricci flow.