Over the last fifty years, the problem of finding sharp upper bounds for area-normalized Laplacian eigenvalues on closed surfaces has attracted the attention of many geometers, due in part to connections to the study of sphere-valued harmonic maps and minimal immersions. In this talk, I'll describe a series of results which shed new light on this problem by relating it to the variational theory of the Dirichlet energy on sphere-valued maps. Recent applications include new (H^{-1}-)stability results for the maximization of the first and second Laplacian eigenvalues, and a proof that metrics maximizing the first Steklov eigenvalue on a surface of genus g and k boundary components limit to the \lambda_1-maximizing metric on the closed surface of genus g as k becomes large (in particular, the associated free boundary minimal surfaces in B^{N+1} converge as varifolds to the associated closed minimal surface in S^N). Based on joint works with Mikhail Karpukhin, Mickael Nahon and Iosif Polterovich.