The Laplacian is a canonical second order elliptic operator that can be defined on any Riemannian manifold. The study of upper bounds for its eigenvalues under the volume constraint is a classical problem of spectral geometry going back to J. Hersch, P. Li and S.-T. Yau. The interest in this problem is partially motivated by a surprising connection to the theory of harmonic maps and minimal surfaces. It turns out that smooth critical points of eigenvalue functionals correspond to metrics induced by a minimal immersion to a sphere. In the present talk we discuss the regularity theorem for maximizers of eigenvalue functionals and survey some recent applications, including optimal isoperimetric inequalities for the sphere and projective plane.