The isoperimetric inequality has a long history in mathematics. In this lecture, we will discuss how the isoperimetric inequality can be generalized to submanifolds in Euclidean space. As a special case, we obtain a sharp isoperimetric inequality for minimal submanifolds of codimension at most 2, answering a question going back to work of Carleman. The proof of that inequality is inspired by, but does not actually use, optimal transport