The Laplacian is a canonical second order elliptic operator 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 particular interest to this problem stems from a surprising connection to the theory of minimal surfaces in spheres. In the present talk we will survey some recent results in the area with an emphasis on the role played by the index of minimal surfaces. In particular, we will discuss some recent applications, including a new lower bounds for the index of minimal spheres as well as the optimal isoperimetric inequality for Laplacian eigenvalues on the projective plane.