One of the first two Fields Medals was awarded to Jesse Douglas for his solution of the Plateau problem; that is, the construction of a surface of least area bounding a general embedded curve in euclidean space. Since 1936 the theory of minimal surfaces of any dimension and codimension has been extensively developed. The theory has also been applied to solve problems in broad areas of geometry and topology. In this lecture we will survey applications of the theory to Riemannian geometry and general relativity.