About fifty years ago, J. Hersch made the simple but striking observation that round metrics maximize the first nonzero eigenvalue of the Laplacian among all geometries of given area on the two-sphere, giving a kind of intrinsic counterpart to classic results of Faber-Krahn and Szego-Weinberger for domains in Euclidean space. In the '90s, Nadirashvili noticed a surprising link between intrinsic eigenvalue optimization problems of this type and minimal surfaces in higher-dimensional spheres, and about 15 years ago, Fraser and Schoen discovered a compelling analog of this story for the Steklov (Dirichlet-to-Neumann) spectrum on surfaces with boundary. In this talk, I'll describe joint work with Misha Karpukhin relating these spectral shape optimization problems to natural variational constructions of harmonic maps, and discuss applications to the existence and regularity of maximizing metrics. Then I'll introduce forthcoming work with Karpukhin, Kusner, and McGrath, in which variants of these techniques are used to produce many new families of minimal surfaces in some classical settings, settling some open problems—and raising some new ones--about the space of minimal surfaces in B^3 and S^3. (To be continued in Peter McGrath's talk on October 2nd.)